405:
9874:
5520:
9905:
416:
43:
531:
481:: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
509:(regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).
681:
5440:
1349:
5188:
9147:
The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre,
9889:
of quadrilaterals is illustrated by the figure to the right. Lower classes are special cases of higher classes they are connected to. Note that "trapezoid" here is referring to the North
American definition (the British equivalent is a trapezium). Inclusive definitions are used throughout.
9526:
one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.
2346:
2094:
8308:
6366:
6537:
5563:
The area of the
Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed
6708:
1064:
3830:
Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named
1801:
2721:
5435:{\displaystyle \det {\begin{bmatrix}0&a^{2}&p^{2}&d^{2}&1\\a^{2}&0&b^{2}&q^{2}&1\\p^{2}&b^{2}&0&c^{2}&1\\d^{2}&q^{2}&c^{2}&0&1\\1&1&1&1&0\end{bmatrix}}=0.}
3836:
Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named
1897:
503:: all four angles are right angles (equiangular). An equivalent condition is that the diagonals bisect each other, and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).
4117:
3982:
2584:
8079:
4410:
7779:
7109:
6982:
4262:
5866:
5727:
7491:
5127:
9018:
2998:
2894:
487:, rhomb: all four sides are of equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly including a square, too).
1540:
3499:
9836:
9716:
The four smaller triangles formed by the diagonals and sides of a convex quadrilateral have the property that the product of the areas of two opposite triangles equals the product of the areas of the other two
4689:
6156:
364:
6060:
7322:
2166:
1931:
8835:
4893:
4533:
8748:
8644:
5952:
3668:
8412:
8115:
7589:
6202:
4424:, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus
1069:
7392:
3372:
3233:
1613:
911:
7226:
6377:
7837:
8906:
6548:
3103:
7887:
9122:
7652:
1344:{\displaystyle {\begin{aligned}K&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd\;}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\,\cos ^{2}{\tfrac {1}{2}}(A+C)}}\end{aligned}}}
601:
Bisect-diagonal quadrilateral: one diagonal bisects the other into equal lengths. Every dart and kite is bisect-diagonal. When both diagonals bisect another, it's a parallelogram.
8554:
1661:
9522:, which connects the midpoints of the diagonals, the segment connecting these points being bisected by the vertex centroid. One more interesting line (in some sense dual to the
5167:
The shape and size of a convex quadrilateral are fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals
1029:
972:
475:
are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.
8477:
1687:
257:
9920:. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as
9738:
2592:
9862:
6858:
6838:
6818:
6798:
6778:
6758:
6738:
783:
of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. They intersect at the "vertex centroid" of the quadrilateral (see
497:(equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree; some define a rhomboid as a parallelogram that is not a rhombus.
225:
205:
185:
165:
9950:
9152:(centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.
1809:
3993:
3861:
2424:
9743:
525:, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.
7941:
4273:
7670:
6993:
6866:
4128:
11287:
5754:
5615:
7399:
4944:
8947:
2902:
2773:
8923:
the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter.
1458:
3437:
11136:
4567:
540:: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.
11432:
10087:
3683:
In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are
10119:
6071:
2341:{\displaystyle K={\tfrac {1}{2}}ab\sin {\alpha }+{\tfrac {1}{4}}{\sqrt {4c^{2}d^{2}-(c^{2}+d^{2}-a^{2}-b^{2}+2ab\cos {\alpha })^{2}}},}
303:
11443:
10668:
Rashid, M. A. & Ajibade, A. O., "Two conditions for a quadrilateral to be cyclic expressed in terms of the lengths of its sides",
5978:
2089:{\displaystyle K={\tfrac {1}{2}}{\sqrt {{\bigl (}(a^{2}+c^{2})-2x^{2}{\bigr )}{\bigl (}(b^{2}+d^{2})-2x^{2}{\bigr )}}}\sin {\varphi }}
11348:"On Some Results Obtained by the Quaternion Analysis Respecting the Inscription of "Gauche" Polygons in Surfaces of the Second Order"
7237:
11452:
8759:
10302:
4796:
655:
In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside the quadrilateral.
5972:
between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence
4430:
8675:
8562:
5877:
3582:
8303:{\displaystyle K\leq {\frac {1}{3+{\sqrt {3}}}}(ab+ac+ad+bc+bd+cd)-{\frac {1}{2(1+{\sqrt {3}})^{2}}}(a^{2}+b^{2}+c^{2}+d^{2})}
11295:
11019:
10286:
8325:
9928:
was also used to mean a skew quadrilateral. A skew quadrilateral together with its diagonals form a (possibly non-regular)
6361:{\displaystyle \sin A+\sin B+\sin C+\sin D=4\sin {\tfrac {1}{2}}(A+B)\,\sin {\tfrac {1}{2}}(A+C)\,\sin {\tfrac {1}{2}}(A+D)}
5578:
The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are
427:
In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral.
7507:
7333:
6532:{\displaystyle {\frac {\tan A\,\tan {B}-\tan C\,\tan D}{\tan A\,\tan C-\tan B\,\tan D}}={\frac {\tan(A+C)}{\tan(A+B)}}.}
5176:
3244:
3111:
11480:
1548:
689:
863:
139:, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g.
11148:
10064:
7166:
5560:
A side of the
Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.
729:(US) or trapezium (Commonwealth): a crossed quadrilateral in which one pair of nonadjacent sides is parallel (like a
10500:
8559:
with equality holding if and only if the diagonals are equal. This follows directly from the quadrilateral identity
6703:{\displaystyle {\frac {\tan A+\tan B+\tan C+\tan D}{\cot A+\cot B+\cot C+\cot D}}=\tan {A}\tan {B}\tan {C}\tan {D}.}
3687:, and if their diagonals have equal length. The list applies to the most general cases, and excludes named subsets.
5557:
Each pair of opposite sides of the
Varignon parallelogram are parallel to a diagonal in the original quadrilateral.
4543:
7794:
10339:
8941:
has the largest area. This is a direct consequence of the fact that the area of a convex quadrilateral satisfies
8866:
3030:
10764:
20:
11386:
9973:- Any quadrilateral can be transformed into another quadrilateral by a projective transformation (homography)
7851:
9056:
726:
7615:
11376:
11347:
2351:
which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle
934:. In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to
11603:
11583:
11397:
11393:
11381:
11035:
9699:
8938:
8507:
1620:
604:
585:
282:
11108:
993:
937:
521:: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into
11578:
11535:
11510:
11401:
9691:
7498:
7116:
1055:
595:
411:
of some types of simple quadrilaterals. (UK) denotes
British English and (US) denotes American English.
11203:"A Set of Rectangles Inscribed in an Orthodiagonal Quadrilateral and Defined by Pascal-Points Circles"
10812:
3527:
794:
of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.
5740:
are the length of the diagonals. The length of the bimedian that connects the midpoints of the sides
1796:{\displaystyle K={\tfrac {1}{4}}\left|\tan \theta \right|\cdot \left|a^{2}+c^{2}-b^{2}-d^{2}\right|.}
1617:
In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to
27:
11638:
11166:
10164:
8855:
5968:
The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance
5545:
The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a
5467:
4554:
1397:
537:
10781:
8441:
230:
12015:
11563:
8487:
7658:
7602:
or degenerate such that one side is equal to the sum of the other three (it has collapsed into a
3027:
suffice for determination of the area, since in any quadrilateral the four values are related by
579:
10900:
Dao Thanh Oai, Leonard
Giugiuc, Problem 12033, American Mathematical Monthly, March 2018, p. 277
2716:{\displaystyle K={\tfrac {1}{4}}{\sqrt {4p^{2}q^{2}-\left(a^{2}+c^{2}-b^{2}-d^{2}\right)^{2}}}.}
11588:
11473:
11420:
10644:
10020:
9965:
9955:
9703:
6161:
Note that the two opposite sides in these formulas are not the two that the bimedian connects.
667:
quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See
576:: a cyclic quadrilateral such that the products of the lengths of the opposing sides are equal.
573:
436:
11311:
Barnett, M. P.; Capitani, J. F. (2006). "Modular chemical geometry and symbolic calculation".
10911:
10046:
11989:
11929:
11568:
11429:
10937:
10094:
9909:
9723:
6165:
5550:
5514:
739:: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a
714:. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two
11164:
David, Fraivert (2019), "Pascal-points quadrilaterals inscribed in a cyclic quadrilateral",
9932:, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite
5574:
The diagonals of the
Varignon parallelogram are the bimedians of the original quadrilateral.
5571:
of the
Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.
2398:
444:
11873:
11643:
11573:
11515:
11440:
11320:
10074:
9960:
9710:
9690:
Let exterior squares be drawn on all sides of a quadrilateral. The segments connecting the
9366:
8931:
8432:
8428:
7599:
5455:
4699:
4558:
561:
557:
543:
10543:
8:
12010:
11979:
11954:
11924:
11919:
11878:
11593:
11424:
11234:
11202:
9945:
9841:
3725:
1892:{\displaystyle K={\tfrac {1}{2}}\left|\tan \theta \right|\cdot \left|a^{2}-b^{2}\right|.}
759:: a special case of a crossed rectangle where two of the sides intersect at right angles.
522:
468:
464:
458:
11410:
11324:
2726:
The first reduces to
Brahmagupta's formula in the cyclic quadrilateral case, since then
11984:
11525:
11255:
11226:
11183:
10768:
10726:
10699:
10574:
10474:
10444:
10259:
10224:
10216:
10181:
9886:
6843:
6823:
6803:
6783:
6763:
6743:
6723:
6183:
The two bimedians are perpendicular if and only if the two diagonals have equal length.
3392:
404:
210:
190:
170:
150:
108:
11227:"Properties of a Pascal points circle in a quadrilateral with perpendicular diagonals"
11081:
10528:
J. L. Coolidge, "A historically interesting formula for the area of a quadrilateral",
10313:
4112:{\displaystyle q={\sqrt {a^{2}+d^{2}-2ad\cos {A}}}={\sqrt {b^{2}+c^{2}-2bc\cos {C}}}.}
1665:
Alternatively, we can write the area in terms of the sides and the intersection angle
493:: a parallelogram in which adjacent sides are of unequal lengths, and some angles are
67:
11964:
11558:
11466:
11291:
11187:
11144:
11015:
10282:
10263:
10228:
10185:
10113:
10060:
9484:
9132:
9128:
8927:
8850:
5962:
4547:
3977:{\displaystyle p={\sqrt {a^{2}+b^{2}-2ab\cos {B}}}={\sqrt {c^{2}+d^{2}-2cd\cos {D}}}}
2579:{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}(ac+bd+pq)(ac+bd-pq)}},}
749:: an antiparallelogram whose sides are two opposite sides and the two diagonals of a
746:
736:
506:
124:
57:
639:
is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.
11493:
11328:
11175:
10401:
10255:
10251:
10208:
10173:
10052:
8074:{\displaystyle K\leq {\tfrac {1}{8}}(a^{2}+b^{2}+c^{2}+d^{2}+p^{2}+q^{2}+pq-ac-bd)}
5579:
5463:
5459:
4542:
is the distance between the midpoints of the diagonals. This is sometimes known as
4405:{\displaystyle q={\sqrt {\frac {(ab+cd)(ac+bd)-2abcd(\cos {A}+\cos {C})}{ad+bc}}}.}
722:, all on the left or all on the right as the figure is traced out) add up to 720°.
632:
is a convex quadrilateral whose four vertices all lie on the perimeter of a square.
551:
286:
10962:
7774:{\displaystyle \displaystyle K\leq {\tfrac {1}{2}}{\sqrt{(ab+cd)(ac+bd)(ad+bc)}}.}
7104:{\displaystyle bc\sin ^{2}{\tfrac {1}{2}}C+ad\cos ^{2}{\tfrac {1}{2}}A=(s-b)(s-c)}
6977:{\displaystyle ad\sin ^{2}{\tfrac {1}{2}}A+bc\cos ^{2}{\tfrac {1}{2}}C=(s-a)(s-d)}
4257:{\displaystyle p={\sqrt {\frac {(ac+bd)(ad+bc)-2abcd(\cos {B}+\cos {D})}{ab+cd}}}}
11959:
11939:
11934:
11904:
11623:
11598:
11530:
11447:
11436:
11281:
10841:"E. A. José García, Two Identities and their Consequences, MATINF, 6 (2020) 5-11"
10070:
9996:
9933:
9168:
4415:
3855:
on each triangle formed by one diagonal and two sides of the quadrilateral. Thus
3759:
668:
664:
518:
432:
275:
267:
120:
53:
10909:
10133:
515:: longer than wide, or wider than long (i.e., a rectangle that is not a square).
11969:
11949:
11914:
11909:
11540:
11520:
10840:
10051:, Undergraduate Texts in Mathematics, Springer-Verlag, Theorem 12.1, page 120,
8920:
8483:
8424:
7595:
6173:
5861:{\displaystyle n={\tfrac {1}{2}}{\sqrt {a^{2}-b^{2}+c^{2}-d^{2}+p^{2}+q^{2}}}.}
5722:{\displaystyle m={\tfrac {1}{2}}{\sqrt {-a^{2}+b^{2}-c^{2}+d^{2}+p^{2}+q^{2}}}}
5478:
5451:
4695:
3852:
294:
271:
263:
95:
91:
10379:
10177:
10159:
10056:
9873:
756:
570:: a kite with two opposite right angles. It is a type of cyclic quadrilateral.
564:. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
12004:
11944:
11795:
11608:
10364:
9878:
9695:
8418:
6177:
5546:
3743:
3684:
3505:
740:
494:
478:
420:
408:
11974:
11844:
11800:
11764:
11754:
11749:
11414:
11405:
10242:
Beauregard, R. A. (2009). "Diametric
Quadrilaterals with Two Equal Sides".
9899:
9518:
Another remarkable line in a convex non-parallelogram quadrilateral is the
9343:
the orthocenters in the same triangles. Then the intersection of the lines
9260:
9136:
9127:
From this inequality it follows that the point inside a quadrilateral that
7603:
773:
719:
396:
Any quadrilateral that is not self-intersecting is a simple quadrilateral.
385:
10751:
Leversha, Gerry, "A property of the diagonals of a cyclic quadrilateral",
10626:
Josefsson, Martin (2016) ‘100.31 Heron-like formulas for quadrilaterals’,
5519:
784:
19:
This article is about four-sided mathematical shapes. For other uses, see
11883:
11790:
11769:
11759:
10912:"An inequality related to the lengths and area of a convex quadrilateral"
9929:
9921:
9904:
9523:
9519:
9264:
7486:{\displaystyle K\leq {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}}
6714:
5179:
5122:{\displaystyle efgh(a+c+b+d)(a+c-b-d)=(agh+cef+beh+dfg)(agh+cef-beh-dfg)}
715:
589:
11453:
The role and function of a hierarchical classification of quadrilaterals
11179:
9230:
respectively. Then the "area centroid" is the intersection of the lines
9142:
9013:{\displaystyle K={\tfrac {1}{2}}pq\sin {\theta }\leq {\tfrac {1}{2}}pq,}
2993:{\displaystyle K={\tfrac {1}{2}}{\sqrt {p^{2}q^{2}-(m^{2}-n^{2})^{2}}}.}
2889:{\displaystyle K={\tfrac {1}{2}}{\sqrt {(m+n+p)(m+n-p)(m+n+q)(m+n-q)}},}
512:
11888:
11744:
11734:
11618:
10220:
9997:"Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram"
9970:
9396:
5458:(that is, the four intersection points of adjacent angle bisectors are
567:
439:): no sides are parallel. (In British English, this was once called a
415:
11332:
11143:. Washington, D.C.: Mathematical Association of America. p. 198.
5601:, the length of the bimedian that connects the midpoints of the sides
1535:{\displaystyle K={\tfrac {1}{2}}ad\sin {A}+{\tfrac {1}{2}}bc\sin {C}.}
11863:
11853:
11830:
11820:
11810:
11739:
11648:
11613:
10727:"A New Formula Concerning the Diagonals and Sides of a Quadrilateral"
9412:
8846:
7229:
5958:
5568:
4122:
Other, more symmetric formulas for the lengths of the diagonals, are
3779:
3707:
3494:{\displaystyle K={\tfrac {1}{2}}|\mathbf {AC} \times \mathbf {BD} |,}
750:
730:
702:
500:
454:
450:
42:
11014:. Mathematical Association of America. pp. 114, 119, 120, 261.
10868:, Wolters–Noordhoff Publishing, The Netherlands, 1969, pp. 129, 132.
10212:
9924:
that contain a "puckered" ring of four atoms. Historically the term
9831:{\displaystyle \cos \theta ={\frac {a^{2}+c^{2}-b^{2}-d^{2}}{2pq}},}
9399:
of a quadrilateral. In a convex quadrilateral, the quasiorthocenter
11868:
11858:
11815:
11774:
11703:
11693:
11683:
11502:
9508:
9395:
of the convex quadrilateral. These points can be used to define an
9271:. But two such points can be constructed in the following way. Let
9268:
9149:
5539:
5535:
4684:{\displaystyle p^{2}q^{2}=a^{2}c^{2}+b^{2}d^{2}-2abcd\cos {(A+C)}.}
4561:, regarding the product of the diagonals in a convex quadrilateral
769:
608:
490:
140:
11458:
2373:
The following two formulas express the area in terms of the sides
1545:
In the case of a cyclic quadrilateral, the latter formula becomes
1416:
Another area formula in terms of the sides and angles, with angle
1058:
expresses the area in terms of the sides and two opposite angles:
270:(self-intersecting, or crossed). Simple quadrilaterals are either
11825:
11805:
11718:
11713:
11708:
11698:
11673:
11628:
11489:
10645:"Diagonals of Quadrilaterals -- Perpendicular, Bisecting or Both"
9709:
For any simple quadrilateral with given edge lengths, there is a
6151:{\displaystyle n={\tfrac {1}{2}}{\sqrt {2(a^{2}+c^{2})-4x^{2}}}.}
3795:
709:
625:
is a quadrilateral with a pair of opposite sides of equal length.
547:
530:
484:
359:{\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ }.}
116:
8435:, into an inequality for a convex quadrilateral. It states that
7900:, with equality if and only if the diagonals are perpendicular.
7394:
with equality only if the diagonals are perpendicular and equal.
6055:{\displaystyle m={\tfrac {1}{2}}{\sqrt {2(b^{2}+d^{2})-4x^{2}}}}
5538:
of the opposite sides. The intersection of the bimedians is the
3678:
3238:
if the lengths of two bimedians and one diagonal are given, and
11633:
11061:
Episodes in Nineteenth and Twentieth Century Euclidean Geometry
8319:
A corollary to Euler's quadrilateral theorem is the inequality
7501:
it directly follows that the area of a quadrilateral satisfies
7325:
3811:
9135:
is the intersection of the diagonals. Hence that point is the
7317:{\displaystyle K\leq {\tfrac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2})}
4416:
Generalizations of the parallelogram law and Ptolemy's theorem
680:
646:
is a quadrilateral with two right angles at opposite vertices.
11678:
10910:
Leonard Mihai Giugiuc; Dao Thanh Oai; Kadir Altintas (2018).
4722:) ≥ −1, it also gives a proof of Ptolemy's inequality.
618:
has two opposite equal sides that when extended, meet at 60°.
472:
10813:"Original Problems Proposed by Stanley Rabinowitz 1963–2005"
9684:
8830:{\displaystyle a^{4}+b^{4}+c^{4}\geq {\tfrac {1}{27}}d^{4}.}
7664:
The area of any quadrilateral also satisfies the inequality
3377:
if the lengths of two diagonals and one bimedian are given.
2105:
is the distance between the midpoints of the diagonals, and
1396:
are two (in fact, any two) opposite angles. This reduces to
785:§ Remarkable points and lines in a convex quadrilateral
461:. Trapezia (UK) and trapezoids (US) include parallelograms.
388:, by repeated rotation around the midpoints of their edges.
8417:
where equality holds if and only if the quadrilateral is a
4888:{\displaystyle XY={\frac {|a^{2}+c^{2}-b^{2}-d^{2}|}{2p}}.}
1806:
In the case of a parallelogram, the latter formula becomes
803:
471:(US): one pair of opposite sides are parallel and the base
77:
4528:{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=p^{2}+q^{2}+4x^{2}}
11012:
Charming Proofs : A Journey Into Elegant Mathematics
10996:
Peter, Thomas, "Maximizing the Area of a Quadrilateral",
8743:{\displaystyle a^{2}+b^{2}+c^{2}>{\tfrac {1}{3}}d^{2}}
8639:{\displaystyle m^{2}+n^{2}={\tfrac {1}{2}}(p^{2}+q^{2}).}
5947:{\displaystyle \displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}
3663:{\displaystyle K={\tfrac {1}{2}}|x_{1}y_{2}-x_{2}y_{1}|.}
2743:
The area can also be expressed in terms of the bimedians
143:). Since "gon" means "angle", it is analogously called a
10967:(from vol. 1, no. 1 to vol. 4, no. 2 known as "Eureka")"
9585:
is the point of intersection of the extensions of sides
5534:
of a quadrilateral are the line segments connecting the
3520:. In two-dimensional Euclidean space, expressing vector
2118:
The last trigonometric area formula including the sides
987:
The area can be also expressed in terms of bimedians as
10445:"Five Proofs of an Area Characterization of Rectangles"
9673:
are called "Pascal points" formed by circle ω on sides
8937:
Of all convex quadrilaterals with given diagonals, the
8926:
The quadrilateral with given side lengths that has the
8407:{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}\geq p^{2}+q^{2}}
3847:
The lengths of the diagonals in a convex quadrilateral
9951:
Perpendicular bisector construction of a quadrilateral
8990:
8958:
8877:
8803:
8719:
8593:
8101:
be the lengths of the sides of a convex quadrilateral
7952:
7919:
be the lengths of the sides of a convex quadrilateral
7862:
7805:
7682:
7410:
7344:
7248:
7177:
7054:
7017:
6927:
6890:
6713:
In the last two formulas, no angle is allowed to be a
6332:
6298:
6264:
6082:
5989:
5765:
5626:
5200:
3593:
3448:
3255:
3122:
2913:
2784:
2603:
2500:
2209:
2177:
1942:
1820:
1698:
1559:
1501:
1469:
1309:
1148:
948:
874:
419:
Convex quadrilaterals by symmetry, represented with a
10486:
Mitchell, Douglas W., "The area of a quadrilateral,"
9844:
9746:
9726:
9694:
of opposite squares are (a) equal in length, and (b)
9155:
The "vertex centroid" is the intersection of the two
9143:
Remarkable points and lines in a convex quadrilateral
9059:
8950:
8915:
is the area of a convex quadrilateral with perimeter
8869:
8762:
8678:
8565:
8510:
8444:
8328:
8118:
7944:
7854:
7797:
7674:
7673:
7618:
7510:
7402:
7336:
7240:
7169:
6996:
6869:
6846:
6826:
6806:
6786:
6766:
6746:
6726:
6551:
6380:
6205:
6074:
5981:
5881:
5880:
5757:
5618:
5582:
and are all bisected by their point of intersection.
5191:
4947:
4799:
4570:
4433:
4276:
4131:
3996:
3864:
3585:
3440:
3247:
3114:
3033:
2905:
2776:
2595:
2427:
2169:
1934:
1812:
1690:
1623:
1551:
1461:
1067:
996:
940:
866:
607:: the four extensions of the sides are tangent to an
306:
233:
213:
193:
173:
153:
127:(vertices). The word is derived from the Latin words
26:"Tetragon" redirects here. For the edible plant, see
11082:"On Two Remarkable Lines Related to a Quadrilateral"
8860:. It is a direct consequence of the area inequality
7584:{\displaystyle K\leq {\sqrt {(s-a)(s-b)(s-c)(s-d)}}}
7132:
If a convex quadrilateral has the consecutive sides
6168:
connection between the bimedians and the diagonals:
857:
The area can be expressed in trigonometric terms as
10700:"Characterizations of Orthodiagonal Quadrilaterals"
10606:
10604:
10602:
10600:
10598:
10596:
753:, hence having one pair of parallel opposite sides.
84:
10199:Jobbings, A. K. (1997). "Quadric Quadrilaterals".
9856:
9830:
9732:
9581:is the point of intersection of the diagonals and
9116:
9012:
8900:
8829:
8742:
8638:
8548:
8486:the quadrilateral is cyclic. This is often called
8471:
8406:
8302:
8073:
7881:
7845:The area of a convex quadrilateral also satisfies
7831:
7773:
7646:
7583:
7485:
7387:{\displaystyle K\leq {\tfrac {1}{4}}(p^{2}+q^{2})}
7386:
7316:
7220:
7103:
6976:
6852:
6832:
6812:
6792:
6772:
6752:
6732:
6702:
6531:
6360:
6164:In a convex quadrilateral, there is the following
6150:
6054:
5946:
5860:
5721:
5434:
5121:
4887:
4683:
4527:
4404:
4256:
4111:
3976:
3662:
3493:
3367:{\displaystyle K={\tfrac {1}{4}}{\sqrt {\cdot }},}
3366:
3228:{\displaystyle K={\tfrac {1}{2}}{\sqrt {\cdot }},}
3227:
3097:
2992:
2888:
2715:
2578:
2340:
2088:
1891:
1795:
1655:
1607:
1534:
1343:
1023:
966:
905:
358:
251:
219:
199:
179:
159:
11253:
10889:, Mathematical Association of America, p. 68
10887:When Less is More: Visualizing Basic Inequalities
10556:Archibald, R. C., "The Area of a Quadrilateral",
10157:
8840:
1608:{\displaystyle K={\tfrac {1}{2}}(ad+bc)\sin {A}.}
797:
12002:
11394:Quadrilaterals Formed by Perpendicular Bisectors
10771:, Geometry Revisited, MAA, 1967, pp. 52–53.
10593:
5192:
4557:derived in 1842 the following generalization of
906:{\displaystyle K={\tfrac {1}{2}}pq\sin \theta ,}
11310:
11000:, Vol. 34, No. 4 (September 2003), pp. 315–316.
9740:at the intersection of the diagonals satisfies
9046:is an interior point in a convex quadrilateral
7221:{\displaystyle K\leq {\tfrac {1}{4}}(a+c)(b+d)}
11003:
9912:represent a regular zig-zag skew quadrilateral
457:(US): at least one pair of opposite sides are
11474:
11106:
10622:
10620:
10568:
10566:
10438:
10436:
10434:
10158:Keady, G.; Scales, P.; Németh, S. Z. (2004).
9186:can be constructed in the following way. Let
3679:Properties of the diagonals in quadrilaterals
2368:
2068:
2016:
2009:
1957:
11430:An extended classification of quadrilaterals
11279:
11141:Euclidean Geometry in Mathematical Olympiads
11075:
11073:
11071:
11069:
11009:
10884:
10693:
10691:
9698:. Thus these centers are the vertices of an
7842:with equality only in the case of a square.
7832:{\displaystyle K\leq {\tfrac {1}{16}}L^{2},}
5466:. In the latter case the quadrilateral is a
1400:for the area of a cyclic quadrilateral—when
147:, or 4-angle. A quadrilateral with vertices
11421:A (dynamic) Hierarchical Quadrilateral Tree
10938:"Properties of equidiagonal quadrilaterals"
9167:coordinates of the vertex centroid are the
8901:{\displaystyle K\leq {\tfrac {1}{16}}L^{2}}
6187:
3842:
3098:{\displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).}
802:There are various general formulas for the
11481:
11467:
11411:Definitions and examples of quadrilaterals
11313:International Journal of Quantum Chemistry
10617:
10563:
10541:
10431:
10377:
10362:
10276:
10241:
9156:
8493:In any convex quadrilateral the bimedians
8314:
7115:We can use these identities to derive the
6192:The four angles of a simple quadrilateral
5531:
2112:
1171:
11218:
11194:
11079:
11066:
10935:
10880:
10878:
10876:
10874:
10697:
10688:
10572:
10442:
9685:Other properties of convex quadrilaterals
6460:
6438:
6417:
6393:
6324:
6290:
4725:
1901:Another area formula including the sides
1294:
852:
675:
650:
11345:
11157:
11036:"Two Centers of Mass of a Quadrilateral"
10992:
10990:
10480:
10198:
9903:
9872:
9259:, there are no natural analogies to the
6800:be the sides of a convex quadrilateral,
5518:
5454:of a convex quadrilateral either form a
4694:This relation can be considered to be a
3431:. The area of the quadrilateral is then
763:
679:
414:
403:
391:
16:Polygon with four sides and four corners
10957:
10955:
10681:Andreescu, Titu & Andrica, Dorian,
10575:"The Area of a Bicentric Quadrilateral"
10424:Harries, J. "Area of a quadrilateral,"
9916:A non-planar quadrilateral is called a
9864:are the diagonals of the quadrilateral.
9633:. Then there holds: the straight lines
8849:, the one with the largest area is the
8109:, then the following inequality holds:
7882:{\displaystyle K\leq {\tfrac {1}{2}}pq}
5965:applied in the Varignon parallelogram.
4934:, and where the diagonals intersect at
1034:where the lengths of the bimedians are
916:where the lengths of the diagonals are
546:: a trapezoid where the four sides are
445:Trapezoid § Trapezium vs Trapezoid
399:
12003:
11415:Definition and properties of tetragons
11355:Proceedings of the Royal Irish Academy
11280:Alsina, Claudi; Nelsen, Roger (2020).
11063:, Math. Assoc. Amer., 1995, pp. 35–41.
11010:Alsina, Claudi; Nelsen, Roger (2010).
10885:Alsina, Claudi; Nelsen, Roger (2009),
10871:
10337:
10118:: CS1 maint: archived copy as title (
10044:
9893:
9877:A taxonomy of quadrilaterals, using a
9117:{\displaystyle AP+BP+CP+DP\geq AC+BD.}
8845:Among all quadrilaterals with a given
5542:of the vertices of the quadrilateral.
5175:of a quadrilateral are related by the
3003:In fact, any three of the four values
135:, meaning "side". It is also called a
11462:
11224:
11200:
11163:
10987:
10929:
10724:
10546:. MathWorld – A Wolfram Web Resource.
10399:
10395:
10393:
10391:
10389:
10382:. MathWorld – A Wolfram Web Resource.
10367:. MathWorld – A Wolfram Web Resource.
9182:The "area centroid" of quadrilateral
7647:{\displaystyle K\leq {\sqrt {abcd}},}
6717:, since tan 90° is not defined.
5585:In a convex quadrilateral with sides
384:All non-self-crossing quadrilaterals
11135:
10952:
10672:, vol. 34 (2003) no. 5, pp. 739–799.
9991:
9989:
9987:
9369:, and the intersection of the lines
6172:The two bimedians have equal length
692:quadrilateral is called variously a
598:: the diagonals are of equal length.
11488:
9027:is the angle between the diagonals
8549:{\displaystyle pq\leq m^{2}+n^{2},}
7493:with equality only for a rectangle.
5553:. It has the following properties:
3504:which is half the magnitude of the
3105:The corresponding expressions are:
2357:), by just changing the first sign
1656:{\displaystyle K=ab\cdot \sin {A}.}
582:: it is both tangential and cyclic.
13:
10386:
10160:"Watt Linkages and Quadrilaterals"
9961:Types of mesh § Quadrilateral
9299:be the circumcenters of triangles
9255:In a general convex quadrilateral
6196:satisfy the following identities:
5445:
3380:
1024:{\displaystyle K=mn\sin \varphi ,}
967:{\displaystyle K={\tfrac {pq}{2}}}
772:of a convex quadrilateral are the
334:
325:
316:
307:
14:
12027:
11369:
11286:. American Mathematical Society.
11256:"Characterizations of Trapezoids"
11207:Journal for Geometry and Graphics
10919:International Journal of Geometry
10670:Int. J. Math. Educ. Sci. Technol.
9984:
9433:as the intersection of the lines
4738:are the feet of the normals from
373:-gon interior angle sum formula:
11346:Hamilton, William Rowan (1850).
10782:"Mateescu Constantin, Answer to
10685:, Birkhäuser, 2006, pp. 207–209.
9035:. Equality holds if and only if
8310:with equality only for a square.
8081:with equality only for a square.
3479:
3476:
3468:
3465:
1354:where the sides in sequence are
776:that connect opposite vertices.
529:
41:
11339:
11304:
11273:
11247:
11129:
11100:
11053:
11028:
10998:The College Mathematics Journal
10903:
10894:
10858:
10833:
10805:
10792:
10774:
10758:
10745:
10718:
10675:
10662:
10637:
10550:
10535:
10522:
10493:
10463:
10418:
10371:
10356:
10331:
10295:
7122:
4546:and is a generalization of the
11441:Dynamic Math Learning Homepage
11283:A Cornucopia of Quadrilaterals
10798:C. V. Durell & A. Robson,
10281:. Springer. pp. 429–430.
10270:
10256:10.1080/07468342.2009.11922331
10235:
10192:
10151:
10126:
10080:
10045:Martin, George Edward (1982),
10038:
10013:
9214:be the centroids of triangles
8841:Maximum and minimum properties
8630:
8604:
8501:are related by the inequality
8297:
8245:
8233:
8216:
8201:
8147:
8068:
7963:
7756:
7738:
7735:
7717:
7714:
7696:
7576:
7564:
7561:
7549:
7546:
7534:
7531:
7519:
7478:
7452:
7449:
7423:
7381:
7355:
7311:
7259:
7215:
7203:
7200:
7188:
7098:
7086:
7083:
7071:
6971:
6959:
6956:
6944:
6520:
6508:
6497:
6485:
6355:
6343:
6321:
6309:
6287:
6275:
6124:
6098:
6031:
6005:
5937:
5911:
5493:, then the angle bisectors of
5116:
5068:
5065:
5017:
5011:
4987:
4984:
4960:
4867:
4813:
4674:
4662:
4375:
4347:
4326:
4308:
4305:
4287:
4230:
4202:
4181:
4163:
4160:
4142:
3653:
3605:
3528:free vector in Cartesian space
3484:
3460:
3356:
3347:
3334:
3315:
3309:
3284:
3271:
3268:
3217:
3208:
3195:
3179:
3173:
3151:
3138:
3135:
3089:
3063:
2976:
2949:
2878:
2860:
2857:
2839:
2836:
2818:
2815:
2797:
2568:
2541:
2538:
2511:
2493:
2481:
2478:
2466:
2463:
2451:
2448:
2436:
2324:
2248:
2047:
2021:
1988:
1962:
1588:
1570:
1332:
1320:
1276:
1264:
1261:
1249:
1246:
1234:
1231:
1219:
1202:
1199:
1187:
1172:
1141:
1129:
1126:
1114:
1111:
1099:
1096:
1084:
1046:and the angle between them is
928:and the angle between them is
798:Area of a convex quadrilateral
369:This is a special case of the
101:90° (for square and rectangle)
21:Quadrilateral (disambiguation)
1:
11109:"Centroid of a quadrilateral"
10683:Complex Numbers from A to...Z
10558:American Mathematical Monthly
10530:American Mathematical Monthly
9977:
9179:coordinates of the vertices.
8669:of any quadrilateral satisfy
4544:Euler's quadrilateral theorem
560:: the four vertices lie on a
10338:Butler, David (2016-04-06).
10021:"Sum of Angles in a Polygon"
9661:that is located on the side
9645:that is located on the side
9593:, let ω be a circle through
9566:pass through a common point
9558:, respectively, the circles
9426:There can also be defined a
9407:, and the quasicircumcenter
9315:respectively; and denote by
9131:the sum of distances to the
8472:{\displaystyle pq\leq ac+bd}
8431:, which is an equality in a
6820:is the semiperimeter, and
5523:The Varignon parallelogram
5508:
4420:In any convex quadrilateral
3851:can be calculated using the
3673:
3576:, this can be rewritten as:
3385:The area of a quadrilateral
266:(not self-intersecting), or
252:{\displaystyle \square ABCD}
47:Some types of quadrilaterals
7:
11382:Encyclopedia of Mathematics
10471:Advanced Euclidean Geometry
10279:Geometry: Euclid and Beyond
10244:College Mathematics Journal
9939:
9868:
9713:with the same edge lengths.
9700:orthodiagonal quadrilateral
9573:For a convex quadrilateral
9159:. As with any polygon, the
9139:of a convex quadrilateral.
8939:orthodiagonal quadrilateral
605:Ex-tangential quadrilateral
586:Orthodiagonal quadrilateral
10:
12032:
11402:Interactive Classification
11254:Josefsson, Martin (2013).
11080:Myakishev, Alexei (2006),
10963:"Inequalities proposed in
10936:Josefsson, Martin (2014).
10698:Josefsson, Martin (2012),
10610:Altshiller-Court, Nathan,
10573:Josefsson, Martin (2011),
10443:Josefsson, Martin (2013),
9897:
7784:Denoting the perimeter as
6860:are opposite angles, then
5512:
5171:and the four side lengths
4898:In a convex quadrilateral
4754:in a convex quadrilateral
4698:for a quadrilateral. In a
2369:Non-trigonometric formulas
1671:of the diagonals, as long
1384:is the semiperimeter, and
811:of a convex quadrilateral
596:Equidiagonal quadrilateral
262:Quadrilaterals are either
25:
18:
11897:
11843:
11783:
11727:
11666:
11657:
11549:
11501:
11425:Dynamic Geometry Sketches
11040:Sites.math.washington.edu
10544:"Bretschneider's formula"
10178:10.1017/S0025557200176107
10057:10.1007/978-1-4612-5680-9
9570:, called a Miquel point.
7324:with equality only for a
7228:with equality only for a
4553:The German mathematician
2111:is the angle between the
588:: the diagonals cross at
431:Irregular quadrilateral (
381:− 2) × 180° (here, n=4).
131:, a variant of four, and
90:
76:
66:
52:
40:
35:
28:Tetragonia tetragonioides
11225:David, Fraivert (2017),
11201:David, Fraivert (2019),
11167:The Mathematical Gazette
10755:93, March 2009, 116–118.
10628:The Mathematical Gazette
10201:The Mathematical Gazette
10165:The Mathematical Gazette
9908:The (red) side edges of
8648:
8482:where there is equality
7606:, so the area is zero).
6188:Trigonometric identities
5468:tangential quadrilateral
4555:Carl Anton Bretschneider
3843:Lengths of the diagonals
3407:form the diagonals from
3391:can be calculated using
538:Tangential quadrilateral
227:is sometimes denoted as
11404:of Quadrilaterals from
11398:Projective Collinearity
10490:93, July 2009, 306–309.
10428:86, July 2002, 310–311.
10340:"The crossed trapezium"
10277:Hartshorne, R. (2005).
10134:"Rectangles Calculator"
10048:Transformation geometry
9733:{\displaystyle \theta }
8315:Diagonals and bimedians
7659:bicentric quadrilateral
7499:Bretschneider's formula
7127:
7117:Bretschneider's Formula
3699:Perpendicular diagonals
1056:Bretschneider's formula
644:Hjelmslev quadrilateral
637:diametric quadrilateral
580:Bicentric quadrilateral
11455:by Michael de Villiers
11377:"Quadrangle, complete"
10866:Geometric Inequalities
10802:, Dover, 2003, p. 267.
10784:Inequality Of Diagonal
10560:, 29 (1922) pp. 29–36.
10303:"Stars: A Second Look"
9966:Quadrangle (geography)
9956:Saccheri quadrilateral
9913:
9882:
9858:
9832:
9734:
9530:For any quadrilateral
9428:quasinine-point center
9403:, the "area centroid"
9118:
9014:
8902:
8831:
8744:
8640:
8550:
8473:
8408:
8304:
8075:
7883:
7833:
7775:
7648:
7585:
7487:
7388:
7318:
7222:
7105:
6978:
6854:
6834:
6814:
6794:
6774:
6754:
6734:
6704:
6533:
6362:
6176:the two diagonals are
6152:
6056:
5948:
5862:
5723:
5551:Varignon parallelogram
5527:
5436:
5123:
4889:
4726:Other metric relations
4710:= 180°, it reduces to
4685:
4529:
4406:
4258:
4113:
3978:
3664:
3495:
3368:
3229:
3099:
2994:
2890:
2717:
2580:
2342:
2090:
1893:
1797:
1657:
1609:
1536:
1345:
1025:
968:
907:
853:Trigonometric formulas
685:
676:Complex quadrilaterals
651:Concave quadrilaterals
574:Harmonic quadrilateral
437:North American English
424:
412:
360:
253:
221:
201:
181:
161:
10800:Advanced Trigonometry
10725:Hoehn, Larry (2011),
10406:mathworld.wolfram.com
10344:Making Your Own Sense
9910:tetragonal disphenoid
9907:
9876:
9859:
9833:
9735:
9649:; the straight lines
9542:the intersections of
9119:
9015:
8903:
8856:isoperimetric theorem
8853:. This is called the
8832:
8745:
8641:
8551:
8474:
8409:
8305:
8076:
7892:for diagonal lengths
7884:
7834:
7776:
7649:
7598:the quadrilateral is
7586:
7488:
7389:
7319:
7223:
7106:
6979:
6855:
6835:
6815:
6795:
6775:
6755:
6735:
6705:
6534:
6363:
6153:
6057:
5949:
5863:
5724:
5522:
5437:
5124:
4890:
4686:
4530:
4407:
4259:
4114:
3979:
3665:
3496:
3369:
3230:
3100:
2995:
2891:
2718:
2581:
2343:
2091:
1894:
1798:
1658:
1610:
1537:
1398:Brahmagupta's formula
1346:
1026:
969:
908:
764:Special line segments
698:crossed quadrilateral
683:
630:quadric quadrilateral
616:equilic quadrilateral
418:
407:
392:Simple quadrilaterals
361:
254:
222:
202:
182:
162:
11714:Nonagon/Enneagon (9)
11644:Tangential trapezoid
10753:Mathematical Gazette
10614:, Dover Publ., 2007.
10532:, 46 (1939) 345–347.
10488:Mathematical Gazette
10426:Mathematical Gazette
9926:gauche quadrilateral
9842:
9744:
9724:
9711:cyclic quadrilateral
9560:(PAB), (PCD), (QAD),
9057:
8948:
8932:cyclic quadrilateral
8867:
8760:
8676:
8563:
8508:
8488:Ptolemy's inequality
8442:
8433:cyclic quadrilateral
8326:
8116:
7942:
7852:
7795:
7671:
7657:with equality for a
7616:
7508:
7400:
7334:
7238:
7167:
6994:
6867:
6844:
6824:
6804:
6784:
6764:
6744:
6724:
6549:
6378:
6203:
6072:
5979:
5878:
5755:
5616:
5456:cyclic quadrilateral
5189:
4945:
4797:
4700:cyclic quadrilateral
4568:
4431:
4274:
4129:
3994:
3862:
3583:
3438:
3245:
3112:
3031:
2903:
2774:
2593:
2425:
2406:, and the diagonals
2167:
1932:
1810:
1688:
1621:
1549:
1459:
1440:being between sides
1422:being between sides
1065:
994:
938:
864:
684:An antiparallelogram
663:(or arrowhead) is a
562:circumscribed circle
558:Cyclic quadrilateral
544:Tangential trapezoid
400:Convex quadrilateral
304:
231:
211:
191:
171:
151:
11826:Megagon (1,000,000)
11594:Isosceles trapezoid
11325:2006IJQC..106..215B
11263:Forum Geometricorum
11235:Forum Geometricorum
11180:10.1017/mag.2019.54
11107:John Boris Miller.
11089:Forum Geometricorum
10965:Crux Mathematicorum
10942:Forum Geometricorum
10734:Forum Geometricorum
10707:Forum Geometricorum
10634:(549), pp. 505–508.
10582:Forum Geometricorum
10501:"Triangle formulae"
10452:Forum Geometricorum
10400:Weisstein, Eric W.
9946:Complete quadrangle
9894:Skew quadrilaterals
9857:{\displaystyle p,q}
9704:Van Aubel's theorem
9657:intersect at point
9641:intersect at point
9503:respectively. Then
9415:in this order, and
3726:Isosceles trapezoid
3696:Bisecting diagonals
694:cross-quadrilateral
523:congruent triangles
469:isosceles trapezoid
465:Isosceles trapezium
11796:Icositetragon (24)
11446:2018-08-25 at the
11435:2019-12-30 at the
11059:Honsberger, Ross,
9918:skew quadrilateral
9914:
9883:
9854:
9828:
9730:
9485:nine-point centers
9114:
9010:
8999:
8967:
8898:
8886:
8858:for quadrilaterals
8827:
8812:
8740:
8728:
8636:
8602:
8546:
8497:and the diagonals
8469:
8404:
8300:
8071:
7961:
7879:
7871:
7829:
7814:
7771:
7770:
7691:
7644:
7581:
7483:
7419:
7384:
7353:
7314:
7257:
7218:
7186:
7148:and the diagonals
7101:
7063:
7026:
6974:
6936:
6899:
6850:
6830:
6810:
6790:
6770:
6750:
6730:
6700:
6529:
6358:
6341:
6307:
6273:
6148:
6091:
6052:
5998:
5944:
5943:
5858:
5774:
5719:
5635:
5528:
5515:Varignon's theorem
5432:
5420:
5119:
4885:
4681:
4525:
4402:
4254:
4109:
3974:
3660:
3602:
3491:
3457:
3364:
3264:
3225:
3131:
3095:
2990:
2922:
2886:
2793:
2755:and the diagonals
2713:
2612:
2576:
2509:
2338:
2218:
2186:
2086:
1951:
1889:
1829:
1793:
1707:
1653:
1605:
1568:
1532:
1510:
1478:
1341:
1339:
1318:
1157:
1021:
964:
962:
903:
883:
686:
623:Watt quadrilateral
425:
413:
356:
249:
217:
197:
177:
157:
11998:
11997:
11839:
11838:
11816:Myriagon (10,000)
11801:Triacontagon (30)
11765:Heptadecagon (17)
11755:Pentadecagon (15)
11750:Tetradecagon (14)
11689:Quadrilateral (4)
11559:Antiparallelogram
11333:10.1002/qua.20807
11297:978-1-47-045312-1
11021:978-0-88385-348-1
10310:Mysite.mweb.co.za
10288:978-1-4419-3145-0
10138:Cleavebooks.co.uk
9823:
9702:. This is called
9367:quasicircumcenter
8998:
8966:
8919:. Equality holds
8885:
8811:
8727:
8601:
8429:Ptolemy's theorem
8427:also generalized
8243:
8230:
8145:
8142:
7960:
7870:
7813:
7765:
7690:
7639:
7579:
7481:
7418:
7352:
7256:
7185:
7062:
7025:
6935:
6898:
6853:{\displaystyle C}
6833:{\displaystyle A}
6813:{\displaystyle s}
6793:{\displaystyle d}
6773:{\displaystyle c}
6753:{\displaystyle b}
6733:{\displaystyle a}
6648:
6524:
6471:
6340:
6306:
6272:
6143:
6090:
6050:
5997:
5963:parallelogram law
5853:
5773:
5717:
5634:
5501:meet on diagonal
5489:meet on diagonal
5473:In quadrilateral
4880:
4559:Ptolemy's theorem
4548:parallelogram law
4397:
4396:
4252:
4251:
4104:
4051:
3972:
3919:
3826:
3825:
3601:
3456:
3359:
3263:
3220:
3130:
2985:
2921:
2881:
2792:
2708:
2611:
2571:
2508:
2333:
2217:
2185:
2073:
1950:
1828:
1706:
1567:
1509:
1477:
1335:
1317:
1205:
1156:
961:
882:
747:Crossed rectangle
737:Antiparallelogram
727:Crossed trapezoid
690:self-intersecting
285:of a simple (and
220:{\displaystyle D}
200:{\displaystyle C}
180:{\displaystyle B}
160:{\displaystyle A}
123:(sides) and four
105:
104:
12023:
11811:Chiliagon (1000)
11791:Icositrigon (23)
11770:Octadecagon (18)
11760:Hexadecagon (16)
11664:
11663:
11483:
11476:
11469:
11460:
11459:
11417:from Mathopenref
11390:
11363:
11362:
11352:
11343:
11337:
11336:
11308:
11302:
11301:
11277:
11271:
11270:
11260:
11251:
11245:
11243:
11231:
11222:
11216:
11214:
11198:
11192:
11190:
11174:(557): 233–239,
11161:
11155:
11154:
11133:
11127:
11126:
11124:
11122:
11113:
11104:
11098:
11096:
11086:
11077:
11064:
11057:
11051:
11050:
11048:
11046:
11032:
11026:
11025:
11007:
11001:
10994:
10985:
10984:
10982:
10980:
10971:
10959:
10950:
10949:
10933:
10927:
10926:
10916:
10907:
10901:
10898:
10892:
10890:
10882:
10869:
10862:
10856:
10855:
10853:
10851:
10837:
10831:
10830:
10828:
10826:
10820:Mathpropress.com
10817:
10809:
10803:
10796:
10790:
10789:
10778:
10772:
10765:H. S. M. Coxeter
10762:
10756:
10749:
10743:
10741:
10731:
10722:
10716:
10714:
10704:
10695:
10686:
10679:
10673:
10666:
10660:
10659:
10657:
10655:
10649:Math.okstate.edu
10641:
10635:
10624:
10615:
10612:College Geometry
10608:
10591:
10589:
10579:
10570:
10561:
10554:
10548:
10547:
10542:E.W. Weisstein.
10539:
10533:
10526:
10520:
10519:
10517:
10515:
10508:mathcentre.ac.uk
10505:
10497:
10491:
10484:
10478:
10467:
10461:
10459:
10449:
10440:
10429:
10422:
10416:
10415:
10413:
10412:
10397:
10384:
10383:
10378:E.W. Weisstein.
10375:
10369:
10368:
10363:E.W. Weisstein.
10360:
10354:
10353:
10351:
10350:
10335:
10329:
10328:
10326:
10324:
10319:on March 3, 2016
10318:
10312:. Archived from
10307:
10299:
10293:
10292:
10274:
10268:
10267:
10239:
10233:
10232:
10207:(491): 220–224.
10196:
10190:
10189:
10172:(513): 475–492.
10155:
10149:
10148:
10146:
10144:
10130:
10124:
10123:
10117:
10109:
10107:
10105:
10099:
10093:. Archived from
10092:
10084:
10078:
10077:
10042:
10036:
10035:
10033:
10031:
10017:
10011:
10010:
10008:
10007:
9993:
9863:
9861:
9860:
9855:
9837:
9835:
9834:
9829:
9824:
9822:
9811:
9810:
9809:
9797:
9796:
9784:
9783:
9771:
9770:
9760:
9739:
9737:
9736:
9731:
9629:meet ω again at
9621:meet ω again at
9393:quasiorthocenter
9169:arithmetic means
9123:
9121:
9120:
9115:
9019:
9017:
9016:
9011:
9000:
8991:
8985:
8968:
8959:
8907:
8905:
8904:
8899:
8897:
8896:
8887:
8878:
8836:
8834:
8833:
8828:
8823:
8822:
8813:
8804:
8798:
8797:
8785:
8784:
8772:
8771:
8749:
8747:
8746:
8741:
8739:
8738:
8729:
8720:
8714:
8713:
8701:
8700:
8688:
8687:
8645:
8643:
8642:
8637:
8629:
8628:
8616:
8615:
8603:
8594:
8588:
8587:
8575:
8574:
8555:
8553:
8552:
8547:
8542:
8541:
8529:
8528:
8478:
8476:
8475:
8470:
8413:
8411:
8410:
8405:
8403:
8402:
8390:
8389:
8377:
8376:
8364:
8363:
8351:
8350:
8338:
8337:
8309:
8307:
8306:
8301:
8296:
8295:
8283:
8282:
8270:
8269:
8257:
8256:
8244:
8242:
8241:
8240:
8231:
8226:
8208:
8146:
8144:
8143:
8138:
8126:
8080:
8078:
8077:
8072:
8040:
8039:
8027:
8026:
8014:
8013:
8001:
8000:
7988:
7987:
7975:
7974:
7962:
7953:
7888:
7886:
7885:
7880:
7872:
7863:
7838:
7836:
7835:
7830:
7825:
7824:
7815:
7806:
7780:
7778:
7777:
7772:
7766:
7764:
7759:
7694:
7692:
7683:
7661:or a rectangle.
7653:
7651:
7650:
7645:
7640:
7626:
7590:
7588:
7587:
7582:
7580:
7518:
7492:
7490:
7489:
7484:
7482:
7477:
7476:
7464:
7463:
7448:
7447:
7435:
7434:
7422:
7420:
7411:
7393:
7391:
7390:
7385:
7380:
7379:
7367:
7366:
7354:
7345:
7323:
7321:
7320:
7315:
7310:
7309:
7297:
7296:
7284:
7283:
7271:
7270:
7258:
7249:
7227:
7225:
7224:
7219:
7187:
7178:
7156:, then its area
7110:
7108:
7107:
7102:
7064:
7055:
7049:
7048:
7027:
7018:
7012:
7011:
6983:
6981:
6980:
6975:
6937:
6928:
6922:
6921:
6900:
6891:
6885:
6884:
6859:
6857:
6856:
6851:
6839:
6837:
6836:
6831:
6819:
6817:
6816:
6811:
6799:
6797:
6796:
6791:
6779:
6777:
6776:
6771:
6759:
6757:
6756:
6751:
6739:
6737:
6736:
6731:
6709:
6707:
6706:
6701:
6696:
6685:
6674:
6663:
6649:
6647:
6600:
6553:
6538:
6536:
6535:
6530:
6525:
6523:
6500:
6477:
6472:
6470:
6427:
6404:
6382:
6367:
6365:
6364:
6359:
6342:
6333:
6308:
6299:
6274:
6265:
6157:
6155:
6154:
6149:
6144:
6142:
6141:
6123:
6122:
6110:
6109:
6094:
6092:
6083:
6061:
6059:
6058:
6053:
6051:
6049:
6048:
6030:
6029:
6017:
6016:
6001:
5999:
5990:
5953:
5951:
5950:
5945:
5936:
5935:
5923:
5922:
5904:
5903:
5891:
5890:
5867:
5865:
5864:
5859:
5854:
5852:
5851:
5839:
5838:
5826:
5825:
5813:
5812:
5800:
5799:
5787:
5786:
5777:
5775:
5766:
5728:
5726:
5725:
5720:
5718:
5716:
5715:
5703:
5702:
5690:
5689:
5677:
5676:
5664:
5663:
5651:
5650:
5638:
5636:
5627:
5441:
5439:
5438:
5433:
5425:
5424:
5380:
5379:
5368:
5367:
5356:
5355:
5337:
5336:
5320:
5319:
5308:
5307:
5289:
5288:
5277:
5276:
5260:
5259:
5241:
5240:
5229:
5228:
5217:
5216:
5128:
5126:
5125:
5120:
4894:
4892:
4891:
4886:
4881:
4879:
4871:
4870:
4865:
4864:
4852:
4851:
4839:
4838:
4826:
4825:
4816:
4810:
4746:to the diagonal
4690:
4688:
4687:
4682:
4677:
4636:
4635:
4626:
4625:
4613:
4612:
4603:
4602:
4590:
4589:
4580:
4579:
4534:
4532:
4531:
4526:
4524:
4523:
4508:
4507:
4495:
4494:
4482:
4481:
4469:
4468:
4456:
4455:
4443:
4442:
4411:
4409:
4408:
4403:
4398:
4395:
4378:
4374:
4360:
4285:
4284:
4263:
4261:
4260:
4255:
4253:
4250:
4233:
4229:
4215:
4140:
4139:
4118:
4116:
4115:
4110:
4105:
4103:
4080:
4079:
4067:
4066:
4057:
4052:
4050:
4027:
4026:
4014:
4013:
4004:
3983:
3981:
3980:
3975:
3973:
3971:
3948:
3947:
3935:
3934:
3925:
3920:
3918:
3895:
3894:
3882:
3881:
3872:
3702:Equal diagonals
3690:
3689:
3669:
3667:
3666:
3661:
3656:
3651:
3650:
3641:
3640:
3628:
3627:
3618:
3617:
3608:
3603:
3594:
3575:
3555:
3549:
3525:
3519:
3513:
3500:
3498:
3497:
3492:
3487:
3482:
3471:
3463:
3458:
3449:
3430:
3424:
3418:
3412:
3406:
3400:
3390:
3373:
3371:
3370:
3365:
3360:
3355:
3354:
3330:
3329:
3308:
3307:
3292:
3291:
3267:
3265:
3256:
3234:
3232:
3231:
3226:
3221:
3216:
3215:
3191:
3190:
3172:
3171:
3159:
3158:
3134:
3132:
3123:
3104:
3102:
3101:
3096:
3088:
3087:
3075:
3074:
3056:
3055:
3043:
3042:
3026:
3020:
3014:
3008:
2999:
2997:
2996:
2991:
2986:
2984:
2983:
2974:
2973:
2961:
2960:
2945:
2944:
2935:
2934:
2925:
2923:
2914:
2895:
2893:
2892:
2887:
2882:
2796:
2794:
2785:
2766:
2760:
2754:
2748:
2739:
2722:
2720:
2719:
2714:
2709:
2707:
2706:
2701:
2697:
2696:
2695:
2683:
2682:
2670:
2669:
2657:
2656:
2638:
2637:
2628:
2627:
2615:
2613:
2604:
2585:
2583:
2582:
2577:
2572:
2510:
2501:
2435:
2417:
2411:
2405:
2396:
2390:
2384:
2378:
2364:
2360:
2356:
2347:
2345:
2344:
2339:
2334:
2332:
2331:
2322:
2299:
2298:
2286:
2285:
2273:
2272:
2260:
2259:
2244:
2243:
2234:
2233:
2221:
2219:
2210:
2204:
2187:
2178:
2159:
2153:
2147:
2141:
2135:
2129:
2123:
2110:
2104:
2095:
2093:
2092:
2087:
2085:
2074:
2072:
2071:
2065:
2064:
2046:
2045:
2033:
2032:
2020:
2019:
2013:
2012:
2006:
2005:
1987:
1986:
1974:
1973:
1961:
1960:
1954:
1952:
1943:
1924:
1918:
1912:
1906:
1898:
1896:
1895:
1890:
1885:
1881:
1880:
1879:
1867:
1866:
1849:
1845:
1830:
1821:
1802:
1800:
1799:
1794:
1789:
1785:
1784:
1783:
1771:
1770:
1758:
1757:
1745:
1744:
1727:
1723:
1708:
1699:
1680:
1676:
1670:
1662:
1660:
1659:
1654:
1649:
1614:
1612:
1611:
1606:
1601:
1569:
1560:
1541:
1539:
1538:
1533:
1528:
1511:
1502:
1496:
1479:
1470:
1451:
1445:
1439:
1433:
1427:
1421:
1412:
1411:
1395:
1389:
1383:
1377:
1371:
1365:
1359:
1350:
1348:
1347:
1342:
1340:
1336:
1319:
1310:
1304:
1303:
1218:
1210:
1206:
1158:
1149:
1083:
1051:
1045:
1039:
1030:
1028:
1027:
1022:
983:
979:
973:
971:
970:
965:
963:
957:
949:
933:
927:
921:
912:
910:
909:
904:
884:
875:
848:
810:
552:inscribed circle
533:
443:. For more, see
435:) or trapezium (
365:
363:
362:
357:
352:
351:
289:) quadrilateral
258:
256:
255:
250:
226:
224:
223:
218:
206:
204:
203:
198:
186:
184:
183:
178:
166:
164:
163:
158:
115:is a four-sided
82:various methods;
72:{4} (for square)
45:
33:
32:
12031:
12030:
12026:
12025:
12024:
12022:
12021:
12020:
12001:
12000:
11999:
11994:
11893:
11847:
11835:
11779:
11745:Tridecagon (13)
11735:Hendecagon (11)
11723:
11659:
11653:
11624:Right trapezoid
11545:
11497:
11487:
11448:Wayback Machine
11437:Wayback Machine
11375:
11372:
11367:
11366:
11350:
11344:
11340:
11309:
11305:
11298:
11278:
11274:
11258:
11252:
11248:
11229:
11223:
11219:
11199:
11195:
11162:
11158:
11151:
11134:
11130:
11120:
11118:
11111:
11105:
11101:
11084:
11078:
11067:
11058:
11054:
11044:
11042:
11034:
11033:
11029:
11022:
11008:
11004:
10995:
10988:
10978:
10976:
10969:
10961:
10960:
10953:
10934:
10930:
10914:
10908:
10904:
10899:
10895:
10883:
10872:
10863:
10859:
10849:
10847:
10839:
10838:
10834:
10824:
10822:
10815:
10811:
10810:
10806:
10797:
10793:
10780:
10779:
10775:
10763:
10759:
10750:
10746:
10729:
10723:
10719:
10702:
10696:
10689:
10680:
10676:
10667:
10663:
10653:
10651:
10643:
10642:
10638:
10625:
10618:
10609:
10594:
10577:
10571:
10564:
10555:
10551:
10540:
10536:
10527:
10523:
10513:
10511:
10503:
10499:
10498:
10494:
10485:
10481:
10469:R. A. Johnson,
10468:
10464:
10447:
10441:
10432:
10423:
10419:
10410:
10408:
10402:"Quadrilateral"
10398:
10387:
10376:
10372:
10361:
10357:
10348:
10346:
10336:
10332:
10322:
10320:
10316:
10305:
10301:
10300:
10296:
10289:
10275:
10271:
10240:
10236:
10213:10.2307/3619199
10197:
10193:
10156:
10152:
10142:
10140:
10132:
10131:
10127:
10111:
10110:
10103:
10101:
10100:on May 14, 2014
10097:
10090:
10088:"Archived copy"
10086:
10085:
10081:
10067:
10043:
10039:
10029:
10027:
10019:
10018:
10014:
10005:
10003:
9995:
9994:
9985:
9980:
9942:
9902:
9896:
9885:A hierarchical
9871:
9843:
9840:
9839:
9812:
9805:
9801:
9792:
9788:
9779:
9775:
9766:
9762:
9761:
9759:
9745:
9742:
9741:
9725:
9722:
9721:
9687:
9481:
9474:
9467:
9460:
9453:
9449:
9442:
9438:
9389:
9385:
9378:
9374:
9363:
9359:
9352:
9348:
9341:
9334:
9327:
9320:
9297:
9290:
9283:
9276:
9250:
9246:
9239:
9235:
9212:
9205:
9198:
9191:
9145:
9058:
9055:
9054:
8989:
8981:
8957:
8949:
8946:
8945:
8892:
8888:
8876:
8868:
8865:
8864:
8843:
8818:
8814:
8802:
8793:
8789:
8780:
8776:
8767:
8763:
8761:
8758:
8757:
8734:
8730:
8718:
8709:
8705:
8696:
8692:
8683:
8679:
8677:
8674:
8673:
8651:
8624:
8620:
8611:
8607:
8592:
8583:
8579:
8570:
8566:
8564:
8561:
8560:
8537:
8533:
8524:
8520:
8509:
8506:
8505:
8443:
8440:
8439:
8398:
8394:
8385:
8381:
8372:
8368:
8359:
8355:
8346:
8342:
8333:
8329:
8327:
8324:
8323:
8317:
8291:
8287:
8278:
8274:
8265:
8261:
8252:
8248:
8236:
8232:
8225:
8212:
8207:
8137:
8130:
8125:
8117:
8114:
8113:
8035:
8031:
8022:
8018:
8009:
8005:
7996:
7992:
7983:
7979:
7970:
7966:
7951:
7943:
7940:
7939:
7861:
7853:
7850:
7849:
7820:
7816:
7804:
7796:
7793:
7792:
7760:
7695:
7693:
7681:
7672:
7669:
7668:
7625:
7617:
7614:
7613:
7517:
7509:
7506:
7505:
7472:
7468:
7459:
7455:
7443:
7439:
7430:
7426:
7421:
7409:
7401:
7398:
7397:
7375:
7371:
7362:
7358:
7343:
7335:
7332:
7331:
7305:
7301:
7292:
7288:
7279:
7275:
7266:
7262:
7247:
7239:
7236:
7235:
7176:
7168:
7165:
7164:
7130:
7125:
7053:
7044:
7040:
7016:
7007:
7003:
6995:
6992:
6991:
6926:
6917:
6913:
6889:
6880:
6876:
6868:
6865:
6864:
6845:
6842:
6841:
6825:
6822:
6821:
6805:
6802:
6801:
6785:
6782:
6781:
6765:
6762:
6761:
6745:
6742:
6741:
6725:
6722:
6721:
6692:
6681:
6670:
6659:
6601:
6554:
6552:
6550:
6547:
6546:
6501:
6478:
6476:
6428:
6400:
6383:
6381:
6379:
6376:
6375:
6331:
6297:
6263:
6204:
6201:
6200:
6190:
6137:
6133:
6118:
6114:
6105:
6101:
6093:
6081:
6073:
6070:
6069:
6044:
6040:
6025:
6021:
6012:
6008:
6000:
5988:
5980:
5977:
5976:
5957:This is also a
5931:
5927:
5918:
5914:
5899:
5895:
5886:
5882:
5879:
5876:
5875:
5847:
5843:
5834:
5830:
5821:
5817:
5808:
5804:
5795:
5791:
5782:
5778:
5776:
5764:
5756:
5753:
5752:
5711:
5707:
5698:
5694:
5685:
5681:
5672:
5668:
5659:
5655:
5646:
5642:
5637:
5625:
5617:
5614:
5613:
5517:
5511:
5479:angle bisectors
5452:angle bisectors
5448:
5446:Angle bisectors
5419:
5418:
5413:
5408:
5403:
5398:
5392:
5391:
5386:
5381:
5375:
5371:
5369:
5363:
5359:
5357:
5351:
5347:
5344:
5343:
5338:
5332:
5328:
5326:
5321:
5315:
5311:
5309:
5303:
5299:
5296:
5295:
5290:
5284:
5280:
5278:
5272:
5268:
5266:
5261:
5255:
5251:
5248:
5247:
5242:
5236:
5232:
5230:
5224:
5220:
5218:
5212:
5208:
5206:
5196:
5195:
5190:
5187:
5186:
4946:
4943:
4942:
4872:
4866:
4860:
4856:
4847:
4843:
4834:
4830:
4821:
4817:
4812:
4811:
4809:
4798:
4795:
4794:
4728:
4661:
4631:
4627:
4621:
4617:
4608:
4604:
4598:
4594:
4585:
4581:
4575:
4571:
4569:
4566:
4565:
4519:
4515:
4503:
4499:
4490:
4486:
4477:
4473:
4464:
4460:
4451:
4447:
4438:
4434:
4432:
4429:
4428:
4418:
4379:
4370:
4356:
4286:
4283:
4275:
4272:
4271:
4234:
4225:
4211:
4141:
4138:
4130:
4127:
4126:
4099:
4075:
4071:
4062:
4058:
4056:
4046:
4022:
4018:
4009:
4005:
4003:
3995:
3992:
3991:
3967:
3943:
3939:
3930:
3926:
3924:
3914:
3890:
3886:
3877:
3873:
3871:
3863:
3860:
3859:
3845:
3837:quadrilateral).
3681:
3676:
3652:
3646:
3642:
3636:
3632:
3623:
3619:
3613:
3609:
3604:
3592:
3584:
3581:
3580:
3572:
3565:
3557:
3551:
3546:
3539:
3531:
3521:
3515:
3509:
3483:
3475:
3464:
3459:
3447:
3439:
3436:
3435:
3426:
3420:
3414:
3408:
3402:
3396:
3386:
3383:
3381:Vector formulas
3350:
3346:
3325:
3321:
3303:
3299:
3287:
3283:
3266:
3254:
3246:
3243:
3242:
3211:
3207:
3186:
3182:
3167:
3163:
3154:
3150:
3133:
3121:
3113:
3110:
3109:
3083:
3079:
3070:
3066:
3051:
3047:
3038:
3034:
3032:
3029:
3028:
3022:
3016:
3010:
3004:
2979:
2975:
2969:
2965:
2956:
2952:
2940:
2936:
2930:
2926:
2924:
2912:
2904:
2901:
2900:
2795:
2783:
2775:
2772:
2771:
2762:
2756:
2750:
2744:
2727:
2702:
2691:
2687:
2678:
2674:
2665:
2661:
2652:
2648:
2647:
2643:
2642:
2633:
2629:
2623:
2619:
2614:
2602:
2594:
2591:
2590:
2499:
2434:
2426:
2423:
2422:
2413:
2407:
2401:
2392:
2386:
2380:
2374:
2371:
2362:
2358:
2352:
2327:
2323:
2318:
2294:
2290:
2281:
2277:
2268:
2264:
2255:
2251:
2239:
2235:
2229:
2225:
2220:
2208:
2200:
2176:
2168:
2165:
2164:
2155:
2149:
2143:
2137:
2131:
2125:
2119:
2106:
2100:
2081:
2067:
2066:
2060:
2056:
2041:
2037:
2028:
2024:
2015:
2014:
2008:
2007:
2001:
1997:
1982:
1978:
1969:
1965:
1956:
1955:
1953:
1941:
1933:
1930:
1929:
1920:
1914:
1908:
1902:
1875:
1871:
1862:
1858:
1857:
1853:
1835:
1831:
1819:
1811:
1808:
1807:
1779:
1775:
1766:
1762:
1753:
1749:
1740:
1736:
1735:
1731:
1713:
1709:
1697:
1689:
1686:
1685:
1678:
1672:
1666:
1645:
1622:
1619:
1618:
1597:
1558:
1550:
1547:
1546:
1524:
1500:
1492:
1468:
1460:
1457:
1456:
1447:
1441:
1435:
1429:
1423:
1417:
1402:
1401:
1391:
1385:
1379:
1373:
1367:
1361:
1355:
1338:
1337:
1308:
1299:
1295:
1217:
1208:
1207:
1147:
1082:
1075:
1068:
1066:
1063:
1062:
1047:
1041:
1035:
995:
992:
991:
981:
975:
950:
947:
939:
936:
935:
929:
923:
917:
873:
865:
862:
861:
855:
816:
806:
800:
766:
678:
653:
433:British English
402:
394:
347:
343:
305:
302:
301:
283:interior angles
232:
229:
228:
212:
209:
208:
192:
189:
188:
172:
169:
168:
152:
149:
148:
83:
68:Schläfli symbol
48:
31:
24:
17:
12:
11:
5:
12029:
12019:
12018:
12016:Quadrilaterals
12013:
11996:
11995:
11993:
11992:
11987:
11982:
11977:
11972:
11967:
11962:
11957:
11952:
11950:Pseudotriangle
11947:
11942:
11937:
11932:
11927:
11922:
11917:
11912:
11907:
11901:
11899:
11895:
11894:
11892:
11891:
11886:
11881:
11876:
11871:
11866:
11861:
11856:
11850:
11848:
11841:
11840:
11837:
11836:
11834:
11833:
11828:
11823:
11818:
11813:
11808:
11803:
11798:
11793:
11787:
11785:
11781:
11780:
11778:
11777:
11772:
11767:
11762:
11757:
11752:
11747:
11742:
11740:Dodecagon (12)
11737:
11731:
11729:
11725:
11724:
11722:
11721:
11716:
11711:
11706:
11701:
11696:
11691:
11686:
11681:
11676:
11670:
11668:
11661:
11655:
11654:
11652:
11651:
11646:
11641:
11636:
11631:
11626:
11621:
11616:
11611:
11606:
11601:
11596:
11591:
11586:
11581:
11576:
11571:
11566:
11561:
11555:
11553:
11551:Quadrilaterals
11547:
11546:
11544:
11543:
11538:
11533:
11528:
11523:
11518:
11513:
11507:
11505:
11499:
11498:
11486:
11485:
11478:
11471:
11463:
11457:
11456:
11450:
11427:
11418:
11408:
11391:
11371:
11370:External links
11368:
11365:
11364:
11338:
11319:(1): 215–227.
11303:
11296:
11288:pp. 17–18
11272:
11246:
11217:
11193:
11156:
11149:
11128:
11099:
11065:
11052:
11027:
11020:
11002:
10986:
10951:
10928:
10902:
10893:
10870:
10857:
10845:Matinf.upit.ro
10832:
10804:
10791:
10773:
10769:S. L. Greitzer
10757:
10744:
10717:
10687:
10674:
10661:
10636:
10616:
10592:
10562:
10549:
10534:
10521:
10492:
10479:
10462:
10430:
10417:
10385:
10370:
10355:
10330:
10294:
10287:
10269:
10234:
10191:
10150:
10125:
10079:
10065:
10037:
10012:
10001:Mathsisfun.com
9982:
9981:
9979:
9976:
9975:
9974:
9968:
9963:
9958:
9953:
9948:
9941:
9938:
9895:
9892:
9870:
9867:
9866:
9865:
9853:
9850:
9847:
9827:
9821:
9818:
9815:
9808:
9804:
9800:
9795:
9791:
9787:
9782:
9778:
9774:
9769:
9765:
9758:
9755:
9752:
9749:
9729:
9718:
9714:
9707:
9686:
9683:
9613:internally at
9605:internally at
9479:
9472:
9465:
9458:
9451:
9447:
9440:
9436:
9391:is called the
9387:
9383:
9376:
9372:
9365:is called the
9361:
9357:
9350:
9346:
9339:
9332:
9325:
9318:
9295:
9288:
9281:
9274:
9248:
9244:
9237:
9233:
9210:
9203:
9196:
9189:
9144:
9141:
9125:
9124:
9113:
9110:
9107:
9104:
9101:
9098:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9021:
9020:
9009:
9006:
9003:
8997:
8994:
8988:
8984:
8980:
8977:
8974:
8971:
8965:
8962:
8956:
8953:
8921:if and only if
8909:
8908:
8895:
8891:
8884:
8881:
8875:
8872:
8842:
8839:
8838:
8837:
8826:
8821:
8817:
8810:
8807:
8801:
8796:
8792:
8788:
8783:
8779:
8775:
8770:
8766:
8751:
8750:
8737:
8733:
8726:
8723:
8717:
8712:
8708:
8704:
8699:
8695:
8691:
8686:
8682:
8650:
8647:
8635:
8632:
8627:
8623:
8619:
8614:
8610:
8606:
8600:
8597:
8591:
8586:
8582:
8578:
8573:
8569:
8557:
8556:
8545:
8540:
8536:
8532:
8527:
8523:
8519:
8516:
8513:
8484:if and only if
8480:
8479:
8468:
8465:
8462:
8459:
8456:
8453:
8450:
8447:
8415:
8414:
8401:
8397:
8393:
8388:
8384:
8380:
8375:
8371:
8367:
8362:
8358:
8354:
8349:
8345:
8341:
8336:
8332:
8316:
8313:
8312:
8311:
8299:
8294:
8290:
8286:
8281:
8277:
8273:
8268:
8264:
8260:
8255:
8251:
8247:
8239:
8235:
8229:
8224:
8221:
8218:
8215:
8211:
8206:
8203:
8200:
8197:
8194:
8191:
8188:
8185:
8182:
8179:
8176:
8173:
8170:
8167:
8164:
8161:
8158:
8155:
8152:
8149:
8141:
8136:
8133:
8129:
8124:
8121:
8105:with the area
8083:
8082:
8070:
8067:
8064:
8061:
8058:
8055:
8052:
8049:
8046:
8043:
8038:
8034:
8030:
8025:
8021:
8017:
8012:
8008:
8004:
7999:
7995:
7991:
7986:
7982:
7978:
7973:
7969:
7965:
7959:
7956:
7950:
7947:
7927:and diagonals
7923:with the area
7890:
7889:
7878:
7875:
7869:
7866:
7860:
7857:
7840:
7839:
7828:
7823:
7819:
7812:
7809:
7803:
7800:
7782:
7781:
7769:
7763:
7758:
7755:
7752:
7749:
7746:
7743:
7740:
7737:
7734:
7731:
7728:
7725:
7722:
7719:
7716:
7713:
7710:
7707:
7704:
7701:
7698:
7689:
7686:
7680:
7677:
7655:
7654:
7643:
7638:
7635:
7632:
7629:
7624:
7621:
7596:if and only if
7594:with equality
7592:
7591:
7578:
7575:
7572:
7569:
7566:
7563:
7560:
7557:
7554:
7551:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7527:
7524:
7521:
7516:
7513:
7495:
7494:
7480:
7475:
7471:
7467:
7462:
7458:
7454:
7451:
7446:
7442:
7438:
7433:
7429:
7425:
7417:
7414:
7408:
7405:
7395:
7383:
7378:
7374:
7370:
7365:
7361:
7357:
7351:
7348:
7342:
7339:
7329:
7313:
7308:
7304:
7300:
7295:
7291:
7287:
7282:
7278:
7274:
7269:
7265:
7261:
7255:
7252:
7246:
7243:
7233:
7217:
7214:
7211:
7208:
7205:
7202:
7199:
7196:
7193:
7190:
7184:
7181:
7175:
7172:
7129:
7126:
7124:
7121:
7113:
7112:
7100:
7097:
7094:
7091:
7088:
7085:
7082:
7079:
7076:
7073:
7070:
7067:
7061:
7058:
7052:
7047:
7043:
7039:
7036:
7033:
7030:
7024:
7021:
7015:
7010:
7006:
7002:
6999:
6985:
6984:
6973:
6970:
6967:
6964:
6961:
6958:
6955:
6952:
6949:
6946:
6943:
6940:
6934:
6931:
6925:
6920:
6916:
6912:
6909:
6906:
6903:
6897:
6894:
6888:
6883:
6879:
6875:
6872:
6849:
6829:
6809:
6789:
6769:
6749:
6729:
6711:
6710:
6699:
6695:
6691:
6688:
6684:
6680:
6677:
6673:
6669:
6666:
6662:
6658:
6655:
6652:
6646:
6643:
6640:
6637:
6634:
6631:
6628:
6625:
6622:
6619:
6616:
6613:
6610:
6607:
6604:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6540:
6539:
6528:
6522:
6519:
6516:
6513:
6510:
6507:
6504:
6499:
6496:
6493:
6490:
6487:
6484:
6481:
6475:
6469:
6466:
6463:
6459:
6456:
6453:
6450:
6447:
6444:
6441:
6437:
6434:
6431:
6426:
6423:
6420:
6416:
6413:
6410:
6407:
6403:
6399:
6396:
6392:
6389:
6386:
6369:
6368:
6357:
6354:
6351:
6348:
6345:
6339:
6336:
6330:
6327:
6323:
6320:
6317:
6314:
6311:
6305:
6302:
6296:
6293:
6289:
6286:
6283:
6280:
6277:
6271:
6268:
6262:
6259:
6256:
6253:
6250:
6247:
6244:
6241:
6238:
6235:
6232:
6229:
6226:
6223:
6220:
6217:
6214:
6211:
6208:
6189:
6186:
6185:
6184:
6181:
6174:if and only if
6159:
6158:
6147:
6140:
6136:
6132:
6129:
6126:
6121:
6117:
6113:
6108:
6104:
6100:
6097:
6089:
6086:
6080:
6077:
6063:
6062:
6047:
6043:
6039:
6036:
6033:
6028:
6024:
6020:
6015:
6011:
6007:
6004:
5996:
5993:
5987:
5984:
5955:
5954:
5942:
5939:
5934:
5930:
5926:
5921:
5917:
5913:
5910:
5907:
5902:
5898:
5894:
5889:
5885:
5869:
5868:
5857:
5850:
5846:
5842:
5837:
5833:
5829:
5824:
5820:
5816:
5811:
5807:
5803:
5798:
5794:
5790:
5785:
5781:
5772:
5769:
5763:
5760:
5730:
5729:
5714:
5710:
5706:
5701:
5697:
5693:
5688:
5684:
5680:
5675:
5671:
5667:
5662:
5658:
5654:
5649:
5645:
5641:
5633:
5630:
5624:
5621:
5576:
5575:
5572:
5565:
5561:
5558:
5510:
5507:
5462:) or they are
5447:
5444:
5443:
5442:
5431:
5428:
5423:
5417:
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5393:
5390:
5387:
5385:
5382:
5378:
5374:
5370:
5366:
5362:
5358:
5354:
5350:
5346:
5345:
5342:
5339:
5335:
5331:
5327:
5325:
5322:
5318:
5314:
5310:
5306:
5302:
5298:
5297:
5294:
5291:
5287:
5283:
5279:
5275:
5271:
5267:
5265:
5262:
5258:
5254:
5250:
5249:
5246:
5243:
5239:
5235:
5231:
5227:
5223:
5219:
5215:
5211:
5207:
5205:
5202:
5201:
5199:
5194:
5182:, as follows:
5130:
5129:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4896:
4895:
4884:
4878:
4875:
4869:
4863:
4859:
4855:
4850:
4846:
4842:
4837:
4833:
4829:
4824:
4820:
4815:
4808:
4805:
4802:
4727:
4724:
4696:law of cosines
4692:
4691:
4680:
4676:
4673:
4670:
4667:
4664:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4634:
4630:
4624:
4620:
4616:
4611:
4607:
4601:
4597:
4593:
4588:
4584:
4578:
4574:
4536:
4535:
4522:
4518:
4514:
4511:
4506:
4502:
4498:
4493:
4489:
4485:
4480:
4476:
4472:
4467:
4463:
4459:
4454:
4450:
4446:
4441:
4437:
4417:
4414:
4413:
4412:
4401:
4394:
4391:
4388:
4385:
4382:
4377:
4373:
4369:
4366:
4363:
4359:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4282:
4279:
4265:
4264:
4249:
4246:
4243:
4240:
4237:
4232:
4228:
4224:
4221:
4218:
4214:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4137:
4134:
4120:
4119:
4108:
4102:
4098:
4095:
4092:
4089:
4086:
4083:
4078:
4074:
4070:
4065:
4061:
4055:
4049:
4045:
4042:
4039:
4036:
4033:
4030:
4025:
4021:
4017:
4012:
4008:
4002:
3999:
3985:
3984:
3970:
3966:
3963:
3960:
3957:
3954:
3951:
3946:
3942:
3938:
3933:
3929:
3923:
3917:
3913:
3910:
3907:
3904:
3901:
3898:
3893:
3889:
3885:
3880:
3876:
3870:
3867:
3853:law of cosines
3844:
3841:
3840:
3839:
3833:
3831:quadrilateral.
3824:
3823:
3820:
3817:
3814:
3808:
3807:
3804:
3801:
3798:
3792:
3791:
3788:
3785:
3782:
3776:
3775:
3770:
3767:
3762:
3756:
3755:
3752:
3749:
3746:
3740:
3739:
3736:
3731:
3728:
3722:
3721:
3718:
3713:
3710:
3704:
3703:
3700:
3697:
3694:
3680:
3677:
3675:
3672:
3671:
3670:
3659:
3655:
3649:
3645:
3639:
3635:
3631:
3626:
3622:
3616:
3612:
3607:
3600:
3597:
3591:
3588:
3570:
3563:
3544:
3537:
3502:
3501:
3490:
3486:
3481:
3478:
3474:
3470:
3467:
3462:
3455:
3452:
3446:
3443:
3395:. Let vectors
3382:
3379:
3375:
3374:
3363:
3358:
3353:
3349:
3345:
3342:
3339:
3336:
3333:
3328:
3324:
3320:
3317:
3314:
3311:
3306:
3302:
3298:
3295:
3290:
3286:
3282:
3279:
3276:
3273:
3270:
3262:
3259:
3253:
3250:
3236:
3235:
3224:
3219:
3214:
3210:
3206:
3203:
3200:
3197:
3194:
3189:
3185:
3181:
3178:
3175:
3170:
3166:
3162:
3157:
3153:
3149:
3146:
3143:
3140:
3137:
3129:
3126:
3120:
3117:
3094:
3091:
3086:
3082:
3078:
3073:
3069:
3065:
3062:
3059:
3054:
3050:
3046:
3041:
3037:
3001:
3000:
2989:
2982:
2978:
2972:
2968:
2964:
2959:
2955:
2951:
2948:
2943:
2939:
2933:
2929:
2920:
2917:
2911:
2908:
2897:
2896:
2885:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2791:
2788:
2782:
2779:
2724:
2723:
2712:
2705:
2700:
2694:
2690:
2686:
2681:
2677:
2673:
2668:
2664:
2660:
2655:
2651:
2646:
2641:
2636:
2632:
2626:
2622:
2618:
2610:
2607:
2601:
2598:
2587:
2586:
2575:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2507:
2504:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2433:
2430:
2370:
2367:
2349:
2348:
2337:
2330:
2326:
2321:
2317:
2314:
2311:
2308:
2305:
2302:
2297:
2293:
2289:
2284:
2280:
2276:
2271:
2267:
2263:
2258:
2254:
2250:
2247:
2242:
2238:
2232:
2228:
2224:
2216:
2213:
2207:
2203:
2199:
2196:
2193:
2190:
2184:
2181:
2175:
2172:
2142:and the angle
2097:
2096:
2084:
2080:
2077:
2070:
2063:
2059:
2055:
2052:
2049:
2044:
2040:
2036:
2031:
2027:
2023:
2018:
2011:
2004:
2000:
1996:
1993:
1990:
1985:
1981:
1977:
1972:
1968:
1964:
1959:
1949:
1946:
1940:
1937:
1888:
1884:
1878:
1874:
1870:
1865:
1861:
1856:
1852:
1848:
1844:
1841:
1838:
1834:
1827:
1824:
1818:
1815:
1804:
1803:
1792:
1788:
1782:
1778:
1774:
1769:
1765:
1761:
1756:
1752:
1748:
1743:
1739:
1734:
1730:
1726:
1722:
1719:
1716:
1712:
1705:
1702:
1696:
1693:
1652:
1648:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1604:
1600:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1566:
1563:
1557:
1554:
1543:
1542:
1531:
1527:
1523:
1520:
1517:
1514:
1508:
1505:
1499:
1495:
1491:
1488:
1485:
1482:
1476:
1473:
1467:
1464:
1352:
1351:
1334:
1331:
1328:
1325:
1322:
1316:
1313:
1307:
1302:
1298:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1216:
1213:
1211:
1209:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1170:
1167:
1164:
1161:
1155:
1152:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1081:
1078:
1076:
1074:
1071:
1070:
1032:
1031:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
960:
956:
953:
946:
943:
914:
913:
902:
899:
896:
893:
890:
887:
881:
878:
872:
869:
854:
851:
799:
796:
765:
762:
761:
760:
757:Crossed square
754:
744:
734:
677:
674:
673:
672:
652:
649:
648:
647:
640:
633:
626:
619:
612:
602:
599:
593:
583:
577:
571:
565:
555:
541:
527:
526:
516:
510:
504:
498:
488:
482:
476:
462:
448:
401:
398:
393:
390:
386:tile the plane
367:
366:
355:
350:
346:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
293:add up to 360
248:
245:
242:
239:
236:
216:
196:
176:
156:
119:, having four
103:
102:
99:
92:Internal angle
88:
87:
80:
74:
73:
70:
64:
63:
60:
50:
49:
46:
38:
37:
15:
9:
6:
4:
3:
2:
12028:
12017:
12014:
12012:
12009:
12008:
12006:
11991:
11990:Weakly simple
11988:
11986:
11983:
11981:
11978:
11976:
11973:
11971:
11968:
11966:
11963:
11961:
11958:
11956:
11953:
11951:
11948:
11946:
11943:
11941:
11938:
11936:
11933:
11931:
11930:Infinite skew
11928:
11926:
11923:
11921:
11918:
11916:
11913:
11911:
11908:
11906:
11903:
11902:
11900:
11896:
11890:
11887:
11885:
11882:
11880:
11877:
11875:
11872:
11870:
11867:
11865:
11862:
11860:
11857:
11855:
11852:
11851:
11849:
11846:
11845:Star polygons
11842:
11832:
11831:Apeirogon (∞)
11829:
11827:
11824:
11822:
11819:
11817:
11814:
11812:
11809:
11807:
11804:
11802:
11799:
11797:
11794:
11792:
11789:
11788:
11786:
11782:
11776:
11775:Icosagon (20)
11773:
11771:
11768:
11766:
11763:
11761:
11758:
11756:
11753:
11751:
11748:
11746:
11743:
11741:
11738:
11736:
11733:
11732:
11730:
11726:
11720:
11717:
11715:
11712:
11710:
11707:
11705:
11702:
11700:
11697:
11695:
11692:
11690:
11687:
11685:
11682:
11680:
11677:
11675:
11672:
11671:
11669:
11665:
11662:
11656:
11650:
11647:
11645:
11642:
11640:
11637:
11635:
11632:
11630:
11627:
11625:
11622:
11620:
11617:
11615:
11612:
11610:
11609:Parallelogram
11607:
11605:
11604:Orthodiagonal
11602:
11600:
11597:
11595:
11592:
11590:
11587:
11585:
11584:Ex-tangential
11582:
11580:
11577:
11575:
11572:
11570:
11567:
11565:
11562:
11560:
11557:
11556:
11554:
11552:
11548:
11542:
11539:
11537:
11534:
11532:
11529:
11527:
11524:
11522:
11519:
11517:
11514:
11512:
11509:
11508:
11506:
11504:
11500:
11495:
11491:
11484:
11479:
11477:
11472:
11470:
11465:
11464:
11461:
11454:
11451:
11449:
11445:
11442:
11438:
11434:
11431:
11428:
11426:
11422:
11419:
11416:
11412:
11409:
11407:
11403:
11399:
11395:
11392:
11388:
11384:
11383:
11378:
11374:
11373:
11360:
11356:
11349:
11342:
11334:
11330:
11326:
11322:
11318:
11314:
11307:
11299:
11293:
11289:
11285:
11284:
11276:
11268:
11264:
11257:
11250:
11241:
11237:
11236:
11228:
11221:
11212:
11208:
11204:
11197:
11189:
11185:
11181:
11177:
11173:
11169:
11168:
11160:
11152:
11150:9780883858394
11146:
11142:
11138:
11132:
11117:
11116:Austmd.org.au
11110:
11103:
11094:
11090:
11083:
11076:
11074:
11072:
11070:
11062:
11056:
11041:
11037:
11031:
11023:
11017:
11013:
11006:
10999:
10993:
10991:
10975:
10968:
10966:
10958:
10956:
10947:
10943:
10939:
10932:
10924:
10920:
10913:
10906:
10897:
10888:
10881:
10879:
10877:
10875:
10867:
10861:
10846:
10842:
10836:
10821:
10814:
10808:
10801:
10795:
10787:
10785:
10777:
10770:
10766:
10761:
10754:
10748:
10739:
10735:
10728:
10721:
10712:
10708:
10701:
10694:
10692:
10684:
10678:
10671:
10665:
10650:
10646:
10640:
10633:
10629:
10623:
10621:
10613:
10607:
10605:
10603:
10601:
10599:
10597:
10587:
10583:
10576:
10569:
10567:
10559:
10553:
10545:
10538:
10531:
10525:
10509:
10502:
10496:
10489:
10483:
10476:
10472:
10466:
10457:
10453:
10446:
10439:
10437:
10435:
10427:
10421:
10407:
10403:
10396:
10394:
10392:
10390:
10381:
10374:
10366:
10359:
10345:
10341:
10334:
10315:
10311:
10304:
10298:
10290:
10284:
10280:
10273:
10265:
10261:
10257:
10253:
10249:
10245:
10238:
10230:
10226:
10222:
10218:
10214:
10210:
10206:
10202:
10195:
10187:
10183:
10179:
10175:
10171:
10167:
10166:
10161:
10154:
10139:
10135:
10129:
10121:
10115:
10096:
10089:
10083:
10076:
10072:
10068:
10066:0-387-90636-3
10062:
10058:
10054:
10050:
10049:
10041:
10026:
10022:
10016:
10002:
9998:
9992:
9990:
9988:
9983:
9972:
9969:
9967:
9964:
9962:
9959:
9957:
9954:
9952:
9949:
9947:
9944:
9943:
9937:
9935:
9931:
9927:
9923:
9919:
9911:
9906:
9901:
9891:
9888:
9880:
9879:Hasse diagram
9875:
9851:
9848:
9845:
9825:
9819:
9816:
9813:
9806:
9802:
9798:
9793:
9789:
9785:
9780:
9776:
9772:
9767:
9763:
9756:
9753:
9750:
9747:
9727:
9719:
9715:
9712:
9708:
9705:
9701:
9697:
9696:perpendicular
9693:
9689:
9688:
9682:
9680:
9676:
9672:
9668:
9664:
9660:
9656:
9652:
9648:
9644:
9640:
9636:
9632:
9628:
9624:
9620:
9616:
9612:
9608:
9604:
9600:
9596:
9592:
9588:
9584:
9580:
9576:
9571:
9569:
9565:
9561:
9557:
9553:
9549:
9545:
9541:
9537:
9533:
9528:
9525:
9521:
9516:
9514:
9510:
9506:
9502:
9498:
9494:
9490:
9487:of triangles
9486:
9482:
9475:
9468:
9461:
9454:
9443:
9432:
9429:
9424:
9422:
9418:
9414:
9410:
9406:
9402:
9398:
9394:
9390:
9379:
9368:
9364:
9353:
9342:
9335:
9328:
9321:
9314:
9310:
9306:
9302:
9298:
9291:
9284:
9277:
9270:
9266:
9262:
9258:
9253:
9251:
9240:
9229:
9225:
9221:
9217:
9213:
9206:
9199:
9192:
9185:
9180:
9178:
9174:
9170:
9166:
9162:
9158:
9153:
9151:
9140:
9138:
9134:
9130:
9111:
9108:
9105:
9102:
9099:
9096:
9093:
9090:
9087:
9084:
9081:
9078:
9075:
9072:
9069:
9066:
9063:
9060:
9053:
9052:
9051:
9049:
9045:
9040:
9038:
9034:
9030:
9026:
9007:
9004:
9001:
8995:
8992:
8986:
8982:
8978:
8975:
8972:
8969:
8963:
8960:
8954:
8951:
8944:
8943:
8942:
8940:
8935:
8933:
8929:
8924:
8922:
8918:
8914:
8893:
8889:
8882:
8879:
8873:
8870:
8863:
8862:
8861:
8859:
8857:
8852:
8848:
8824:
8819:
8815:
8808:
8805:
8799:
8794:
8790:
8786:
8781:
8777:
8773:
8768:
8764:
8756:
8755:
8754:
8735:
8731:
8724:
8721:
8715:
8710:
8706:
8702:
8697:
8693:
8689:
8684:
8680:
8672:
8671:
8670:
8668:
8664:
8660:
8656:
8646:
8633:
8625:
8621:
8617:
8612:
8608:
8598:
8595:
8589:
8584:
8580:
8576:
8571:
8567:
8543:
8538:
8534:
8530:
8525:
8521:
8517:
8514:
8511:
8504:
8503:
8502:
8500:
8496:
8491:
8489:
8485:
8466:
8463:
8460:
8457:
8454:
8451:
8448:
8445:
8438:
8437:
8436:
8434:
8430:
8426:
8422:
8420:
8419:parallelogram
8399:
8395:
8391:
8386:
8382:
8378:
8373:
8369:
8365:
8360:
8356:
8352:
8347:
8343:
8339:
8334:
8330:
8322:
8321:
8320:
8292:
8288:
8284:
8279:
8275:
8271:
8266:
8262:
8258:
8253:
8249:
8237:
8227:
8222:
8219:
8213:
8209:
8204:
8198:
8195:
8192:
8189:
8186:
8183:
8180:
8177:
8174:
8171:
8168:
8165:
8162:
8159:
8156:
8153:
8150:
8139:
8134:
8131:
8127:
8122:
8119:
8112:
8111:
8110:
8108:
8104:
8100:
8096:
8092:
8088:
8065:
8062:
8059:
8056:
8053:
8050:
8047:
8044:
8041:
8036:
8032:
8028:
8023:
8019:
8015:
8010:
8006:
8002:
7997:
7993:
7989:
7984:
7980:
7976:
7971:
7967:
7957:
7954:
7948:
7945:
7938:
7937:
7936:
7934:
7930:
7926:
7922:
7918:
7914:
7910:
7906:
7901:
7899:
7895:
7876:
7873:
7867:
7864:
7858:
7855:
7848:
7847:
7846:
7843:
7826:
7821:
7817:
7810:
7807:
7801:
7798:
7791:
7790:
7789:
7787:
7767:
7761:
7753:
7750:
7747:
7744:
7741:
7732:
7729:
7726:
7723:
7720:
7711:
7708:
7705:
7702:
7699:
7687:
7684:
7678:
7675:
7667:
7666:
7665:
7662:
7660:
7641:
7636:
7633:
7630:
7627:
7622:
7619:
7612:
7611:
7610:
7607:
7605:
7601:
7597:
7573:
7570:
7567:
7558:
7555:
7552:
7543:
7540:
7537:
7528:
7525:
7522:
7514:
7511:
7504:
7503:
7502:
7500:
7473:
7469:
7465:
7460:
7456:
7444:
7440:
7436:
7431:
7427:
7415:
7412:
7406:
7403:
7396:
7376:
7372:
7368:
7363:
7359:
7349:
7346:
7340:
7337:
7330:
7327:
7306:
7302:
7298:
7293:
7289:
7285:
7280:
7276:
7272:
7267:
7263:
7253:
7250:
7244:
7241:
7234:
7231:
7212:
7209:
7206:
7197:
7194:
7191:
7182:
7179:
7173:
7170:
7163:
7162:
7161:
7159:
7155:
7151:
7147:
7143:
7139:
7135:
7120:
7118:
7095:
7092:
7089:
7080:
7077:
7074:
7068:
7065:
7059:
7056:
7050:
7045:
7041:
7037:
7034:
7031:
7028:
7022:
7019:
7013:
7008:
7004:
7000:
6997:
6990:
6989:
6988:
6968:
6965:
6962:
6953:
6950:
6947:
6941:
6938:
6932:
6929:
6923:
6918:
6914:
6910:
6907:
6904:
6901:
6895:
6892:
6886:
6881:
6877:
6873:
6870:
6863:
6862:
6861:
6847:
6827:
6807:
6787:
6767:
6747:
6727:
6718:
6716:
6697:
6693:
6689:
6686:
6682:
6678:
6675:
6671:
6667:
6664:
6660:
6656:
6653:
6650:
6644:
6641:
6638:
6635:
6632:
6629:
6626:
6623:
6620:
6617:
6614:
6611:
6608:
6605:
6602:
6597:
6594:
6591:
6588:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6555:
6545:
6544:
6543:
6526:
6517:
6514:
6511:
6505:
6502:
6494:
6491:
6488:
6482:
6479:
6473:
6467:
6464:
6461:
6457:
6454:
6451:
6448:
6445:
6442:
6439:
6435:
6432:
6429:
6424:
6421:
6418:
6414:
6411:
6408:
6405:
6401:
6397:
6394:
6390:
6387:
6384:
6374:
6373:
6372:
6352:
6349:
6346:
6337:
6334:
6328:
6325:
6318:
6315:
6312:
6303:
6300:
6294:
6291:
6284:
6281:
6278:
6269:
6266:
6260:
6257:
6254:
6251:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6209:
6206:
6199:
6198:
6197:
6195:
6182:
6179:
6178:perpendicular
6175:
6171:
6170:
6169:
6167:
6162:
6145:
6138:
6134:
6130:
6127:
6119:
6115:
6111:
6106:
6102:
6095:
6087:
6084:
6078:
6075:
6068:
6067:
6066:
6045:
6041:
6037:
6034:
6026:
6022:
6018:
6013:
6009:
6002:
5994:
5991:
5985:
5982:
5975:
5974:
5973:
5971:
5966:
5964:
5960:
5940:
5932:
5928:
5924:
5919:
5915:
5908:
5905:
5900:
5896:
5892:
5887:
5883:
5874:
5873:
5872:
5855:
5848:
5844:
5840:
5835:
5831:
5827:
5822:
5818:
5814:
5809:
5805:
5801:
5796:
5792:
5788:
5783:
5779:
5770:
5767:
5761:
5758:
5751:
5750:
5749:
5747:
5743:
5739:
5735:
5712:
5708:
5704:
5699:
5695:
5691:
5686:
5682:
5678:
5673:
5669:
5665:
5660:
5656:
5652:
5647:
5643:
5639:
5631:
5628:
5622:
5619:
5612:
5611:
5610:
5608:
5604:
5600:
5596:
5592:
5588:
5583:
5581:
5573:
5570:
5566:
5562:
5559:
5556:
5555:
5554:
5552:
5548:
5547:parallelogram
5543:
5541:
5537:
5533:
5526:
5521:
5516:
5506:
5504:
5500:
5496:
5492:
5488:
5484:
5480:
5476:
5471:
5469:
5465:
5461:
5457:
5453:
5450:The internal
5429:
5426:
5421:
5415:
5410:
5405:
5400:
5395:
5388:
5383:
5376:
5372:
5364:
5360:
5352:
5348:
5340:
5333:
5329:
5323:
5316:
5312:
5304:
5300:
5292:
5285:
5281:
5273:
5269:
5263:
5256:
5252:
5244:
5237:
5233:
5225:
5221:
5213:
5209:
5203:
5197:
5185:
5184:
5183:
5181:
5178:
5177:Cayley-Menger
5174:
5170:
5165:
5163:
5159:
5155:
5151:
5147:
5143:
5139:
5135:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5074:
5071:
5062:
5059:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5029:
5026:
5023:
5020:
5014:
5008:
5005:
5002:
4999:
4996:
4993:
4990:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4957:
4954:
4951:
4948:
4941:
4940:
4939:
4937:
4933:
4929:
4925:
4921:
4917:
4913:
4909:
4905:
4901:
4882:
4876:
4873:
4861:
4857:
4853:
4848:
4844:
4840:
4835:
4831:
4827:
4822:
4818:
4806:
4803:
4800:
4793:
4792:
4791:
4789:
4785:
4781:
4777:
4773:
4769:
4765:
4761:
4757:
4753:
4749:
4745:
4741:
4737:
4733:
4723:
4721:
4717:
4714:. Since cos (
4713:
4709:
4705:
4701:
4697:
4678:
4671:
4668:
4665:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4632:
4628:
4622:
4618:
4614:
4609:
4605:
4599:
4595:
4591:
4586:
4582:
4576:
4572:
4564:
4563:
4562:
4560:
4556:
4551:
4549:
4545:
4541:
4520:
4516:
4512:
4509:
4504:
4500:
4496:
4491:
4487:
4483:
4478:
4474:
4470:
4465:
4461:
4457:
4452:
4448:
4444:
4439:
4435:
4427:
4426:
4425:
4423:
4399:
4392:
4389:
4386:
4383:
4380:
4371:
4367:
4364:
4361:
4357:
4353:
4350:
4344:
4341:
4338:
4335:
4332:
4329:
4323:
4320:
4317:
4314:
4311:
4302:
4299:
4296:
4293:
4290:
4280:
4277:
4270:
4269:
4268:
4247:
4244:
4241:
4238:
4235:
4226:
4222:
4219:
4216:
4212:
4208:
4205:
4199:
4196:
4193:
4190:
4187:
4184:
4178:
4175:
4172:
4169:
4166:
4157:
4154:
4151:
4148:
4145:
4135:
4132:
4125:
4124:
4123:
4106:
4100:
4096:
4093:
4090:
4087:
4084:
4081:
4076:
4072:
4068:
4063:
4059:
4053:
4047:
4043:
4040:
4037:
4034:
4031:
4028:
4023:
4019:
4015:
4010:
4006:
4000:
3997:
3990:
3989:
3988:
3968:
3964:
3961:
3958:
3955:
3952:
3949:
3944:
3940:
3936:
3931:
3927:
3921:
3915:
3911:
3908:
3905:
3902:
3899:
3896:
3891:
3887:
3883:
3878:
3874:
3868:
3865:
3858:
3857:
3856:
3854:
3850:
3838:
3834:
3832:
3828:
3827:
3821:
3818:
3815:
3813:
3810:
3809:
3805:
3802:
3799:
3797:
3794:
3793:
3789:
3786:
3783:
3781:
3778:
3777:
3774:
3771:
3768:
3766:
3763:
3761:
3758:
3757:
3753:
3750:
3747:
3745:
3744:Parallelogram
3742:
3741:
3737:
3735:
3732:
3729:
3727:
3724:
3723:
3719:
3717:
3714:
3711:
3709:
3706:
3705:
3701:
3698:
3695:
3693:Quadrilateral
3692:
3691:
3688:
3686:
3685:perpendicular
3657:
3647:
3643:
3637:
3633:
3629:
3624:
3620:
3614:
3610:
3598:
3595:
3589:
3586:
3579:
3578:
3577:
3573:
3569:
3562:
3554:
3547:
3543:
3536:
3529:
3524:
3518:
3512:
3507:
3506:cross product
3488:
3472:
3453:
3450:
3444:
3441:
3434:
3433:
3432:
3429:
3423:
3417:
3411:
3405:
3399:
3394:
3389:
3378:
3361:
3351:
3343:
3340:
3337:
3331:
3326:
3322:
3318:
3312:
3304:
3300:
3296:
3293:
3288:
3280:
3277:
3274:
3260:
3257:
3251:
3248:
3241:
3240:
3239:
3222:
3212:
3204:
3201:
3198:
3192:
3187:
3183:
3176:
3168:
3164:
3160:
3155:
3147:
3144:
3141:
3127:
3124:
3118:
3115:
3108:
3107:
3106:
3092:
3084:
3080:
3076:
3071:
3067:
3060:
3057:
3052:
3048:
3044:
3039:
3035:
3025:
3019:
3013:
3007:
2987:
2980:
2970:
2966:
2962:
2957:
2953:
2946:
2941:
2937:
2931:
2927:
2918:
2915:
2909:
2906:
2899:
2898:
2883:
2875:
2872:
2869:
2866:
2863:
2854:
2851:
2848:
2845:
2842:
2833:
2830:
2827:
2824:
2821:
2812:
2809:
2806:
2803:
2800:
2789:
2786:
2780:
2777:
2770:
2769:
2768:
2765:
2759:
2753:
2747:
2741:
2738:
2734:
2730:
2710:
2703:
2698:
2692:
2688:
2684:
2679:
2675:
2671:
2666:
2662:
2658:
2653:
2649:
2644:
2639:
2634:
2630:
2624:
2620:
2616:
2608:
2605:
2599:
2596:
2589:
2588:
2573:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2505:
2502:
2496:
2490:
2487:
2484:
2475:
2472:
2469:
2460:
2457:
2454:
2445:
2442:
2439:
2431:
2428:
2421:
2420:
2419:
2416:
2410:
2404:
2400:
2399:semiperimeter
2395:
2389:
2383:
2377:
2366:
2355:
2335:
2328:
2319:
2315:
2312:
2309:
2306:
2303:
2300:
2295:
2291:
2287:
2282:
2278:
2274:
2269:
2265:
2261:
2256:
2252:
2245:
2240:
2236:
2230:
2226:
2222:
2214:
2211:
2205:
2201:
2197:
2194:
2191:
2188:
2182:
2179:
2173:
2170:
2163:
2162:
2161:
2158:
2152:
2146:
2140:
2134:
2128:
2122:
2116:
2114:
2109:
2103:
2082:
2078:
2075:
2061:
2057:
2053:
2050:
2042:
2038:
2034:
2029:
2025:
2002:
1998:
1994:
1991:
1983:
1979:
1975:
1970:
1966:
1947:
1944:
1938:
1935:
1928:
1927:
1926:
1923:
1917:
1911:
1905:
1899:
1886:
1882:
1876:
1872:
1868:
1863:
1859:
1854:
1850:
1846:
1842:
1839:
1836:
1832:
1825:
1822:
1816:
1813:
1790:
1786:
1780:
1776:
1772:
1767:
1763:
1759:
1754:
1750:
1746:
1741:
1737:
1732:
1728:
1724:
1720:
1717:
1714:
1710:
1703:
1700:
1694:
1691:
1684:
1683:
1682:
1675:
1669:
1663:
1650:
1646:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1615:
1602:
1598:
1594:
1591:
1585:
1582:
1579:
1576:
1573:
1564:
1561:
1555:
1552:
1529:
1525:
1521:
1518:
1515:
1512:
1506:
1503:
1497:
1493:
1489:
1486:
1483:
1480:
1474:
1471:
1465:
1462:
1455:
1454:
1453:
1450:
1444:
1438:
1432:
1426:
1420:
1414:
1409:
1405:
1399:
1394:
1388:
1382:
1376:
1370:
1364:
1358:
1329:
1326:
1323:
1314:
1311:
1305:
1300:
1296:
1291:
1288:
1285:
1282:
1279:
1273:
1270:
1267:
1258:
1255:
1252:
1243:
1240:
1237:
1228:
1225:
1222:
1214:
1212:
1196:
1193:
1190:
1184:
1181:
1178:
1175:
1168:
1165:
1162:
1159:
1153:
1150:
1144:
1138:
1135:
1132:
1123:
1120:
1117:
1108:
1105:
1102:
1093:
1090:
1087:
1079:
1077:
1072:
1061:
1060:
1059:
1057:
1053:
1050:
1044:
1038:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
990:
989:
988:
985:
978:
958:
954:
951:
944:
941:
932:
926:
920:
900:
897:
894:
891:
888:
885:
879:
876:
870:
867:
860:
859:
858:
850:
847:
843:
839:
835:
831:
827:
823:
819:
814:
809:
805:
795:
793:
788:
786:
782:
777:
775:
774:line segments
771:
758:
755:
752:
748:
745:
742:
741:parallelogram
738:
735:
732:
728:
725:
724:
723:
721:
717:
713:
712:quadrilateral
711:
706:
705:quadrilateral
704:
699:
695:
691:
682:
670:
666:
662:
658:
657:
656:
645:
641:
638:
634:
631:
627:
624:
620:
617:
613:
610:
606:
603:
600:
597:
594:
591:
587:
584:
581:
578:
575:
572:
569:
566:
563:
559:
556:
553:
549:
545:
542:
539:
536:
535:
534:
532:
524:
520:
517:
514:
511:
508:
505:
502:
499:
496:
492:
489:
486:
483:
480:
479:Parallelogram
477:
474:
470:
466:
463:
460:
456:
452:
449:
446:
442:
438:
434:
430:
429:
428:
422:
421:Hasse diagram
417:
410:
409:Euler diagram
406:
397:
389:
387:
382:
380:
376:
372:
353:
348:
344:
340:
337:
331:
328:
322:
319:
313:
310:
300:
299:
298:
296:
292:
288:
284:
279:
277:
273:
269:
265:
260:
246:
243:
240:
237:
234:
214:
194:
174:
154:
146:
142:
138:
134:
130:
126:
122:
118:
114:
113:quadrilateral
110:
100:
97:
93:
89:
86:
81:
79:
75:
71:
69:
65:
61:
59:
55:
51:
44:
39:
36:Quadrilateral
34:
29:
22:
11784:>20 sides
11719:Decagon (10)
11704:Heptagon (7)
11694:Pentagon (5)
11688:
11684:Triangle (3)
11579:Equidiagonal
11550:
11406:cut-the-knot
11380:
11358:
11354:
11341:
11316:
11312:
11306:
11282:
11275:
11266:
11262:
11249:
11239:
11233:
11220:
11210:
11206:
11196:
11171:
11165:
11159:
11140:
11131:
11119:. Retrieved
11115:
11102:
11092:
11088:
11060:
11055:
11043:. Retrieved
11039:
11030:
11011:
11005:
10997:
10977:. Retrieved
10973:
10964:
10945:
10941:
10931:
10922:
10918:
10905:
10896:
10886:
10865:
10864:O. Bottema,
10860:
10848:. Retrieved
10844:
10835:
10823:. Retrieved
10819:
10807:
10799:
10794:
10783:
10776:
10760:
10752:
10747:
10737:
10733:
10720:
10710:
10706:
10682:
10677:
10669:
10664:
10652:. Retrieved
10648:
10639:
10631:
10627:
10611:
10585:
10581:
10557:
10552:
10537:
10529:
10524:
10512:. Retrieved
10507:
10495:
10487:
10482:
10470:
10465:
10455:
10451:
10425:
10420:
10409:. Retrieved
10405:
10373:
10358:
10347:. Retrieved
10343:
10333:
10321:. Retrieved
10314:the original
10309:
10297:
10278:
10272:
10250:(1): 17–21.
10247:
10243:
10237:
10204:
10200:
10194:
10169:
10163:
10153:
10141:. Retrieved
10137:
10128:
10102:. Retrieved
10095:the original
10082:
10047:
10040:
10028:. Retrieved
10024:
10015:
10004:. Retrieved
10000:
9936:is removed.
9925:
9917:
9915:
9900:Skew polygon
9884:
9678:
9674:
9670:
9666:
9662:
9658:
9654:
9650:
9646:
9642:
9638:
9634:
9630:
9626:
9622:
9618:
9614:
9610:
9606:
9602:
9601:which meets
9598:
9594:
9590:
9586:
9582:
9578:
9574:
9572:
9567:
9563:
9559:
9555:
9551:
9547:
9543:
9539:
9535:
9534:with points
9531:
9529:
9517:
9512:
9504:
9500:
9496:
9492:
9488:
9477:
9470:
9463:
9456:
9445:
9434:
9430:
9427:
9425:
9420:
9416:
9408:
9404:
9400:
9392:
9381:
9370:
9355:
9344:
9337:
9330:
9323:
9316:
9312:
9308:
9304:
9300:
9293:
9286:
9279:
9272:
9261:circumcenter
9256:
9254:
9242:
9231:
9227:
9223:
9219:
9215:
9208:
9201:
9194:
9187:
9183:
9181:
9176:
9172:
9164:
9160:
9154:
9148:called just
9146:
9137:Fermat point
9126:
9047:
9043:
9041:
9036:
9032:
9028:
9024:
9022:
8936:
8930:area is the
8925:
8916:
8912:
8910:
8854:
8844:
8752:
8666:
8662:
8658:
8654:
8652:
8558:
8498:
8494:
8492:
8481:
8423:
8416:
8318:
8106:
8102:
8098:
8094:
8090:
8086:
8084:
7932:
7928:
7924:
7920:
7916:
7912:
7908:
7904:
7902:
7897:
7893:
7891:
7844:
7841:
7785:
7783:
7663:
7656:
7608:
7604:line segment
7593:
7496:
7157:
7153:
7149:
7145:
7141:
7137:
7133:
7131:
7123:Inequalities
7114:
6986:
6719:
6712:
6541:
6370:
6193:
6191:
6163:
6160:
6064:
5969:
5967:
5956:
5870:
5745:
5741:
5737:
5733:
5731:
5606:
5602:
5598:
5594:
5590:
5586:
5584:
5577:
5544:
5529:
5524:
5502:
5498:
5494:
5490:
5486:
5482:
5474:
5472:
5449:
5172:
5168:
5166:
5161:
5157:
5153:
5149:
5145:
5141:
5137:
5133:
5131:
4935:
4931:
4927:
4923:
4919:
4915:
4911:
4907:
4903:
4899:
4897:
4787:
4783:
4779:
4775:
4771:
4767:
4763:
4759:
4755:
4751:
4747:
4743:
4739:
4735:
4731:
4729:
4719:
4715:
4712:pq = ac + bd
4711:
4707:
4703:
4693:
4552:
4539:
4537:
4421:
4419:
4266:
4121:
3986:
3848:
3846:
3835:
3829:
3772:
3764:
3733:
3715:
3682:
3567:
3560:
3559:
3552:
3541:
3534:
3533:
3522:
3516:
3510:
3503:
3427:
3421:
3415:
3409:
3403:
3397:
3387:
3384:
3376:
3237:
3023:
3017:
3011:
3005:
3002:
2763:
2757:
2751:
2745:
2742:
2736:
2732:
2728:
2725:
2414:
2408:
2402:
2393:
2387:
2381:
2375:
2372:
2353:
2350:
2156:
2150:
2144:
2138:
2132:
2126:
2120:
2117:
2107:
2101:
2098:
1921:
1915:
1909:
1903:
1900:
1805:
1673:
1667:
1664:
1616:
1544:
1448:
1442:
1436:
1430:
1424:
1418:
1415:
1407:
1403:
1392:
1386:
1380:
1374:
1368:
1362:
1356:
1353:
1054:
1048:
1042:
1036:
1033:
986:
976:
930:
924:
918:
915:
856:
845:
841:
837:
833:
829:
825:
821:
817:
812:
807:
801:
791:
789:
780:
778:
767:
708:
701:
697:
693:
687:
660:
654:
643:
636:
629:
622:
615:
590:right angles
528:
440:
426:
395:
383:
378:
374:
370:
368:
290:
280:
261:
144:
136:
132:
128:
112:
106:
11980:Star-shaped
11955:Rectilinear
11925:Equilateral
11920:Equiangular
11884:Hendecagram
11728:11–20 sides
11709:Octagon (8)
11699:Hexagon (6)
11674:Monogon (1)
11516:Equilateral
10974:Imomath.com
10475:Dover Publ.
10380:"Maltitude"
9930:tetrahedron
9922:cyclobutane
9520:Newton line
9265:orthocenter
6715:right angle
5549:called the
5180:determinant
4902:with sides
4758:with sides
3508:of vectors
815:with sides
12011:4 (number)
12005:Categories
11985:Tangential
11889:Dodecagram
11667:1–10 sides
11658:By number
11639:Tangential
11619:Right kite
11361:: 380–387.
11137:Chen, Evan
10948:: 129–144.
10411:2020-09-02
10365:"Bimedian"
10349:2017-09-13
10006:2020-09-02
9978:References
9971:Homography
9898:See also:
9720:The angle
9717:triangles.
9665:. Points
9397:Euler line
8653:The sides
7788:, we have
7160:satisfies
5580:concurrent
5513:See also:
5464:concurrent
5173:a, b, c, d
3773:See note 2
3765:See note 2
3734:See note 1
3716:See note 1
792:maltitudes
568:Right kite
297:, that is
145:quadrangle
11965:Reinhardt
11874:Enneagram
11864:Heptagram
11854:Pentagram
11821:65537-gon
11679:Digon (2)
11649:Trapezoid
11614:Rectangle
11564:Bicentric
11526:Isosceles
11503:Triangles
11387:EMS Press
11242:: 509–526
11188:233360695
11095:: 289–295
10740:: 211–212
10588:: 155–164
10264:122206817
10229:250440553
10186:125102050
9799:−
9786:−
9754:θ
9751:
9728:θ
9577:in which
9413:collinear
9157:bimedians
9129:minimizes
9094:≥
8987:≤
8983:θ
8979:
8874:≤
8847:perimeter
8800:≥
8518:≤
8452:≤
8379:≥
8205:−
8123:≤
8060:−
8051:−
7949:≤
7859:≤
7802:≤
7679:≤
7623:≤
7571:−
7556:−
7541:−
7526:−
7515:≤
7407:≤
7341:≤
7245:≤
7230:rectangle
7174:≤
7093:−
7078:−
7051:
7014:
6966:−
6951:−
6924:
6887:
6690:
6679:
6668:
6657:
6642:
6630:
6618:
6606:
6595:
6583:
6571:
6559:
6506:
6483:
6465:
6455:
6449:−
6443:
6433:
6422:
6412:
6406:−
6398:
6388:
6329:
6295:
6261:
6246:
6234:
6222:
6210:
6128:−
6035:−
5959:corollary
5815:−
5789:−
5666:−
5640:−
5569:perimeter
5536:midpoints
5532:bimedians
5509:Bimedians
5477:, if the
5460:concyclic
5105:−
5093:−
5006:−
5000:−
4854:−
4841:−
4659:
4638:−
4368:
4354:
4330:−
4223:
4209:
4185:−
4097:
4082:−
4044:
4029:−
3965:
3950:−
3912:
3897:−
3780:Rectangle
3708:Trapezoid
3674:Diagonals
3630:−
3530:equal to
3473:×
3419:and from
3341:−
3332:−
3313:⋅
3294:−
3202:−
3193:−
3177:⋅
3161:−
2963:−
2947:−
2873:−
2831:−
2685:−
2672:−
2640:−
2560:−
2497:−
2488:−
2473:−
2458:−
2443:−
2320:α
2316:
2288:−
2275:−
2246:−
2202:α
2198:
2148:(between
2113:bimedians
2083:φ
2079:
2051:−
1992:−
1869:−
1851:⋅
1843:θ
1840:
1773:−
1760:−
1729:⋅
1721:θ
1718:
1643:
1637:⋅
1595:
1522:
1490:
1306:
1280:−
1271:−
1256:−
1241:−
1226:−
1185:
1145:−
1136:−
1121:−
1106:−
1091:−
1016:φ
1013:
898:θ
895:
790:The four
781:bimedians
770:diagonals
751:rectangle
731:trapezoid
703:butterfly
501:Rectangle
455:trapezoid
451:Trapezium
441:trapezoid
349:∘
335:∠
326:∠
317:∠
308:∠
235:◻
85:see below
11940:Isotoxal
11935:Isogonal
11879:Decagram
11869:Octagram
11859:Hexagram
11660:of sides
11589:Harmonic
11490:Polygons
11444:Archived
11433:Archived
11269:: 23–35.
11139:(2016).
11121:March 1,
10979:March 1,
10925:: 81–86.
10825:March 1,
10477:, p. 82.
10473:, 2007,
10323:March 1,
10114:cite web
10104:June 20,
9940:See also
9887:taxonomy
9869:Taxonomy
9625:and let
9524:Newton's
9509:midpoint
9483:are the
9455:, where
9269:triangle
9150:centroid
9133:vertices
5540:centroid
4702:, where
1378:, where
787:below).
779:The two
768:The two
718:and two
609:excircle
548:tangents
491:Rhomboid
467:(UK) or
459:parallel
453:(UK) or
141:pentagon
137:tetragon
109:geometry
58:vertices
11960:Regular
11905:Concave
11898:Classes
11806:257-gon
11629:Rhombus
11569:Crossed
11389:, 2001
11321:Bibcode
11045:1 March
10850:1 March
10713:: 13–25
10654:1 March
10514:26 June
10458:: 17–21
10221:3619199
10143:1 March
10075:0718119
10030:22 June
10025:Cuemath
9692:centers
9507:is the
9171:of the
9050:, then
9039:= 90°.
8928:maximum
7935:. Then
5961:to the
4790:, then
3796:Rhombus
3393:vectors
1677:is not
710:bow-tie
665:concave
495:oblique
485:Rhombus
295:degrees
276:concave
268:complex
125:corners
117:polygon
96:degrees
11970:Simple
11915:Cyclic
11910:Convex
11634:Square
11574:Cyclic
11536:Obtuse
11531:Kepler
11294:
11213:: 5–27
11186:
11147:
11018:
10510:. 2009
10285:
10262:
10227:
10219:
10184:
10073:
10063:
9838:where
9617:. Let
9023:where
8911:where
8851:square
8665:, and
7933:BD = q
7929:AC = p
7609:Also,
7600:cyclic
7326:square
6542:Also,
5871:Hence
5732:where
5156:, and
5132:where
4538:where
3812:Square
3021:, and
2397:, the
2160:) is:
2099:where
1434:, and
1410:= 180°
974:since
720:reflex
550:to an
513:Oblong
507:Square
473:angles
287:planar
272:convex
264:simple
129:quadri
11945:Magic
11541:Right
11521:Ideal
11511:Acute
11400:and
11351:(PDF)
11259:(PDF)
11230:(PDF)
11184:S2CID
11112:(PDF)
11085:(PDF)
10970:(PDF)
10915:(PDF)
10816:(PDF)
10730:(PDF)
10703:(PDF)
10578:(PDF)
10504:(PDF)
10448:(PDF)
10317:(PDF)
10306:(PDF)
10260:S2CID
10225:S2CID
10217:JSTOR
10182:S2CID
10098:(PDF)
10091:(PDF)
9934:edges
9564:(QBC)
9267:of a
8649:Sides
8425:Euler
7497:From
3526:as a
1452:, is
716:acute
133:latus
121:edges
54:Edges
11975:Skew
11599:Kite
11494:List
11413:and
11292:ISBN
11145:ISBN
11123:2022
11047:2022
11016:ISBN
10981:2022
10852:2022
10827:2022
10767:and
10656:2022
10516:2023
10325:2022
10283:ISBN
10145:2022
10120:link
10106:2013
10061:ISBN
10032:2022
9677:and
9669:and
9653:and
9637:and
9609:and
9597:and
9589:and
9575:ABCD
9562:and
9554:and
9550:and
9546:and
9538:and
9532:ABCD
9444:and
9411:are
9380:and
9354:and
9263:and
9257:ABCD
9241:and
9184:ABCD
9175:and
9163:and
9048:ABCD
9031:and
8753:and
8716:>
8499:p, q
8495:m, n
8103:ABCD
8085:Let
7921:ABCD
7903:Let
7896:and
7128:Area
6987:and
6840:and
6720:Let
6371:and
6194:ABCD
6166:dual
6065:and
5744:and
5736:and
5605:and
5597:and
5567:The
5530:The
5525:EFGH
5497:and
5485:and
5475:ABCD
5169:p, q
4900:ABCD
4756:ABCD
4742:and
4734:and
4422:ABCD
4267:and
3987:and
3849:ABCD
3822:Yes
3790:Yes
3760:Kite
3738:Yes
3550:and
3514:and
3401:and
3388:ABCD
2391:and
2154:and
1446:and
1428:and
1390:and
1040:and
922:and
840:and
813:ABCD
804:area
669:Kite
661:dart
519:Kite
291:ABCD
281:The
207:and
107:In
78:Area
56:and
11439:at
11423:at
11329:doi
11317:106
11176:doi
11172:103
10632:100
10252:doi
10209:doi
10174:doi
10053:doi
9748:cos
9511:of
9501:ABC
9497:ABD
9493:ACD
9489:BCD
9419:= 2
9313:ABC
9309:ABD
9305:ACD
9301:BCD
9228:ABC
9224:ABD
9220:ACD
9216:BCD
9042:If
8976:sin
7042:cos
7005:sin
6915:cos
6878:sin
6687:tan
6676:tan
6665:tan
6654:tan
6639:cot
6627:cot
6615:cot
6603:cot
6592:tan
6580:tan
6568:tan
6556:tan
6503:tan
6480:tan
6462:tan
6452:tan
6440:tan
6430:tan
6419:tan
6409:tan
6395:tan
6385:tan
6326:sin
6292:sin
6258:sin
6243:sin
6231:sin
6219:sin
6207:sin
5748:is
5609:is
5564:of.
5481:of
5193:det
4730:If
4656:cos
4365:cos
4351:cos
4220:cos
4206:cos
4094:cos
4041:cos
3962:cos
3909:cos
3819:Yes
3816:Yes
3806:No
3803:Yes
3800:Yes
3784:Yes
3769:Yes
3754:No
3748:Yes
3720:No
3556:as
3425:to
3413:to
2361:to
2313:cos
2195:sin
2076:sin
1925:is
1837:tan
1715:tan
1679:90°
1640:sin
1592:sin
1519:sin
1487:sin
1297:cos
1182:cos
1010:sin
982:90°
980:is
892:sin
820:=
707:or
614:An
377:= (
345:360
274:or
12007::
11396:,
11385:,
11379:,
11357:.
11353:.
11327:.
11315:.
11290:.
11267:13
11265:.
11261:.
11240:17
11238:,
11232:,
11211:23
11209:,
11205:,
11182:,
11170:,
11114:.
11091:,
11087:,
11068:^
11038:.
10989:^
10972:.
10954:^
10946:14
10944:.
10940:.
10921:.
10917:.
10873:^
10843:.
10818:.
10738:11
10736:,
10732:,
10711:12
10709:,
10705:,
10690:^
10647:.
10630:,
10619:^
10595:^
10586:11
10584:,
10580:,
10565:^
10506:.
10456:13
10454:,
10450:,
10433:^
10404:.
10388:^
10342:.
10308:.
10258:.
10248:40
10246:.
10223:.
10215:.
10205:81
10203:.
10180:.
10170:88
10168:.
10162:.
10136:.
10116:}}
10112:{{
10071:MR
10069:,
10059:,
10023:.
9999:.
9986:^
9681:.
9679:CD
9675:AB
9663:CD
9655:KM
9651:NL
9647:AB
9639:ML
9635:NK
9627:DB
9619:CA
9611:DA
9603:CB
9591:AD
9587:BC
9556:CD
9552:AB
9548:BC
9544:AD
9515:.
9513:OH
9499:,
9495:,
9491:,
9476:,
9469:,
9462:,
9423:.
9421:GO
9417:HG
9336:,
9329:,
9322:,
9311:,
9307:,
9303:,
9292:,
9285:,
9278:,
9252:.
9226:,
9222:,
9218:,
9207:,
9200:,
9193:,
8934:.
8883:16
8809:27
8661:,
8657:,
8490:.
8421:.
8097:,
8093:,
8089:,
7931:,
7915:,
7911:,
7907:,
7811:16
7152:,
7144:,
7140:,
7136:,
7119:.
6780:,
6760:,
6740:,
5593:,
5589:,
5505:.
5503:AC
5491:BD
5470:.
5430:0.
5164:.
5162:DE
5160:=
5154:CE
5152:=
5148:,
5146:BE
5144:=
5140:,
5138:AE
5136:=
4938:,
4932:DA
4930:=
4926:,
4924:CD
4922:=
4918:,
4916:BC
4914:=
4910:,
4908:AB
4906:=
4788:DA
4786:=
4782:,
4780:CD
4778:=
4774:,
4772:BC
4770:=
4766:,
4764:AB
4762:=
4750:=
4748:AC
4718:+
4706:+
4550:.
3787:No
3751:No
3730:No
3712:No
3553:BD
3523:AC
3517:BD
3511:AC
3404:BD
3398:AC
3015:,
3009:,
2767::
2761:,
2749:,
2740:.
2737:bd
2735:+
2733:ac
2731:=
2729:pq
2418::
2412:,
2385:,
2379:,
2365:.
2136:,
2130:,
2124:,
2115:.
1919:,
1913:,
1907:,
1681::
1413:.
1406:+
1372:,
1366:,
1360:,
1052:.
984:.
849:.
846:DA
844:=
838:CD
836:=
832:,
830:BC
828:=
824:,
822:AB
743:).
733:).
700:,
696:,
688:A
659:A
642:A
635:A
628:A
621:A
447:.)
278:.
259:.
187:,
167:,
111:a
11496:)
11492:(
11482:e
11475:t
11468:v
11359:4
11335:.
11331::
11323::
11300:.
11244:.
11215:.
11191:.
11178::
11153:.
11125:.
11097:.
11093:6
11049:.
11024:.
10983:.
10923:7
10891:.
10854:.
10829:.
10788:.
10786:"
10742:.
10715:.
10658:.
10590:.
10518:.
10460:.
10414:.
10352:.
10327:.
10291:.
10266:.
10254::
10231:.
10211::
10188:.
10176::
10147:.
10122:)
10108:.
10055::
10034:.
10009:.
9881:.
9852:q
9849:,
9846:p
9826:,
9820:q
9817:p
9814:2
9807:2
9803:d
9794:2
9790:b
9781:2
9777:c
9773:+
9768:2
9764:a
9757:=
9706:.
9671:Q
9667:P
9659:Q
9643:P
9631:K
9623:L
9615:N
9607:M
9599:F
9595:E
9583:F
9579:E
9568:M
9540:Q
9536:P
9505:E
9480:d
9478:E
9473:c
9471:E
9466:b
9464:E
9459:a
9457:E
9452:d
9450:E
9448:b
9446:E
9441:c
9439:E
9437:a
9435:E
9431:E
9409:O
9405:G
9401:H
9388:d
9386:H
9384:b
9382:H
9377:c
9375:H
9373:a
9371:H
9362:d
9360:O
9358:b
9356:O
9351:c
9349:O
9347:a
9345:O
9340:d
9338:H
9333:c
9331:H
9326:b
9324:H
9319:a
9317:H
9296:d
9294:O
9289:c
9287:O
9282:b
9280:O
9275:a
9273:O
9249:d
9247:G
9245:b
9243:G
9238:c
9236:G
9234:a
9232:G
9211:d
9209:G
9204:c
9202:G
9197:b
9195:G
9190:a
9188:G
9177:y
9173:x
9165:y
9161:x
9112:.
9109:D
9106:B
9103:+
9100:C
9097:A
9091:P
9088:D
9085:+
9082:P
9079:C
9076:+
9073:P
9070:B
9067:+
9064:P
9061:A
9044:P
9037:θ
9033:q
9029:p
9025:θ
9008:,
9005:q
9002:p
8996:2
8993:1
8973:q
8970:p
8964:2
8961:1
8955:=
8952:K
8917:L
8913:K
8894:2
8890:L
8880:1
8871:K
8825:.
8820:4
8816:d
8806:1
8795:4
8791:c
8787:+
8782:4
8778:b
8774:+
8769:4
8765:a
8736:2
8732:d
8725:3
8722:1
8711:2
8707:c
8703:+
8698:2
8694:b
8690:+
8685:2
8681:a
8667:d
8663:c
8659:b
8655:a
8634:.
8631:)
8626:2
8622:q
8618:+
8613:2
8609:p
8605:(
8599:2
8596:1
8590:=
8585:2
8581:n
8577:+
8572:2
8568:m
8544:,
8539:2
8535:n
8531:+
8526:2
8522:m
8515:q
8512:p
8467:d
8464:b
8461:+
8458:c
8455:a
8449:q
8446:p
8400:2
8396:q
8392:+
8387:2
8383:p
8374:2
8370:d
8366:+
8361:2
8357:c
8353:+
8348:2
8344:b
8340:+
8335:2
8331:a
8298:)
8293:2
8289:d
8285:+
8280:2
8276:c
8272:+
8267:2
8263:b
8259:+
8254:2
8250:a
8246:(
8238:2
8234:)
8228:3
8223:+
8220:1
8217:(
8214:2
8210:1
8202:)
8199:d
8196:c
8193:+
8190:d
8187:b
8184:+
8181:c
8178:b
8175:+
8172:d
8169:a
8166:+
8163:c
8160:a
8157:+
8154:b
8151:a
8148:(
8140:3
8135:+
8132:3
8128:1
8120:K
8107:K
8099:d
8095:c
8091:b
8087:a
8069:)
8066:d
8063:b
8057:c
8054:a
8048:q
8045:p
8042:+
8037:2
8033:q
8029:+
8024:2
8020:p
8016:+
8011:2
8007:d
8003:+
7998:2
7994:c
7990:+
7985:2
7981:b
7977:+
7972:2
7968:a
7964:(
7958:8
7955:1
7946:K
7925:K
7917:d
7913:c
7909:b
7905:a
7898:q
7894:p
7877:q
7874:p
7868:2
7865:1
7856:K
7827:,
7822:2
7818:L
7808:1
7799:K
7786:L
7768:.
7762:3
7757:)
7754:c
7751:b
7748:+
7745:d
7742:a
7739:(
7736:)
7733:d
7730:b
7727:+
7724:c
7721:a
7718:(
7715:)
7712:d
7709:c
7706:+
7703:b
7700:a
7697:(
7688:2
7685:1
7676:K
7642:,
7637:d
7634:c
7631:b
7628:a
7620:K
7577:)
7574:d
7568:s
7565:(
7562:)
7559:c
7553:s
7550:(
7547:)
7544:b
7538:s
7535:(
7532:)
7529:a
7523:s
7520:(
7512:K
7479:)
7474:2
7470:d
7466:+
7461:2
7457:b
7453:(
7450:)
7445:2
7441:c
7437:+
7432:2
7428:a
7424:(
7416:2
7413:1
7404:K
7382:)
7377:2
7373:q
7369:+
7364:2
7360:p
7356:(
7350:4
7347:1
7338:K
7328:.
7312:)
7307:2
7303:d
7299:+
7294:2
7290:c
7286:+
7281:2
7277:b
7273:+
7268:2
7264:a
7260:(
7254:4
7251:1
7242:K
7232:.
7216:)
7213:d
7210:+
7207:b
7204:(
7201:)
7198:c
7195:+
7192:a
7189:(
7183:4
7180:1
7171:K
7158:K
7154:q
7150:p
7146:d
7142:c
7138:b
7134:a
7111:.
7099:)
7096:c
7090:s
7087:(
7084:)
7081:b
7075:s
7072:(
7069:=
7066:A
7060:2
7057:1
7046:2
7038:d
7035:a
7032:+
7029:C
7023:2
7020:1
7009:2
7001:c
6998:b
6972:)
6969:d
6963:s
6960:(
6957:)
6954:a
6948:s
6945:(
6942:=
6939:C
6933:2
6930:1
6919:2
6911:c
6908:b
6905:+
6902:A
6896:2
6893:1
6882:2
6874:d
6871:a
6848:C
6828:A
6808:s
6788:d
6768:c
6748:b
6728:a
6698:.
6694:D
6683:C
6672:B
6661:A
6651:=
6645:D
6636:+
6633:C
6624:+
6621:B
6612:+
6609:A
6598:D
6589:+
6586:C
6577:+
6574:B
6565:+
6562:A
6527:.
6521:)
6518:B
6515:+
6512:A
6509:(
6498:)
6495:C
6492:+
6489:A
6486:(
6474:=
6468:D
6458:B
6446:C
6436:A
6425:D
6415:C
6402:B
6391:A
6356:)
6353:D
6350:+
6347:A
6344:(
6338:2
6335:1
6322:)
6319:C
6316:+
6313:A
6310:(
6304:2
6301:1
6288:)
6285:B
6282:+
6279:A
6276:(
6270:2
6267:1
6255:4
6252:=
6249:D
6240:+
6237:C
6228:+
6225:B
6216:+
6213:A
6180:.
6146:.
6139:2
6135:x
6131:4
6125:)
6120:2
6116:c
6112:+
6107:2
6103:a
6099:(
6096:2
6088:2
6085:1
6079:=
6076:n
6046:2
6042:x
6038:4
6032:)
6027:2
6023:d
6019:+
6014:2
6010:b
6006:(
6003:2
5995:2
5992:1
5986:=
5983:m
5970:x
5941:.
5938:)
5933:2
5929:n
5925:+
5920:2
5916:m
5912:(
5909:2
5906:=
5901:2
5897:q
5893:+
5888:2
5884:p
5856:.
5849:2
5845:q
5841:+
5836:2
5832:p
5828:+
5823:2
5819:d
5810:2
5806:c
5802:+
5797:2
5793:b
5784:2
5780:a
5771:2
5768:1
5762:=
5759:n
5746:d
5742:b
5738:q
5734:p
5713:2
5709:q
5705:+
5700:2
5696:p
5692:+
5687:2
5683:d
5679:+
5674:2
5670:c
5661:2
5657:b
5653:+
5648:2
5644:a
5632:2
5629:1
5623:=
5620:m
5607:c
5603:a
5599:d
5595:c
5591:b
5587:a
5499:D
5495:B
5487:C
5483:A
5427:=
5422:]
5416:0
5411:1
5406:1
5401:1
5396:1
5389:1
5384:0
5377:2
5373:c
5365:2
5361:q
5353:2
5349:d
5341:1
5334:2
5330:c
5324:0
5317:2
5313:b
5305:2
5301:p
5293:1
5286:2
5282:q
5274:2
5270:b
5264:0
5257:2
5253:a
5245:1
5238:2
5234:d
5226:2
5222:p
5214:2
5210:a
5204:0
5198:[
5158:h
5150:g
5142:f
5134:e
5117:)
5114:g
5111:f
5108:d
5102:h
5099:e
5096:b
5090:f
5087:e
5084:c
5081:+
5078:h
5075:g
5072:a
5069:(
5066:)
5063:g
5060:f
5057:d
5054:+
5051:h
5048:e
5045:b
5042:+
5039:f
5036:e
5033:c
5030:+
5027:h
5024:g
5021:a
5018:(
5015:=
5012:)
5009:d
5003:b
4997:c
4994:+
4991:a
4988:(
4985:)
4982:d
4979:+
4976:b
4973:+
4970:c
4967:+
4964:a
4961:(
4958:h
4955:g
4952:f
4949:e
4936:E
4928:d
4920:c
4912:b
4904:a
4883:.
4877:p
4874:2
4868:|
4862:2
4858:d
4849:2
4845:b
4836:2
4832:c
4828:+
4823:2
4819:a
4814:|
4807:=
4804:Y
4801:X
4784:d
4776:c
4768:b
4760:a
4752:p
4744:D
4740:B
4736:Y
4732:X
4720:C
4716:A
4708:C
4704:A
4679:.
4675:)
4672:C
4669:+
4666:A
4663:(
4653:d
4650:c
4647:b
4644:a
4641:2
4633:2
4629:d
4623:2
4619:b
4615:+
4610:2
4606:c
4600:2
4596:a
4592:=
4587:2
4583:q
4577:2
4573:p
4540:x
4521:2
4517:x
4513:4
4510:+
4505:2
4501:q
4497:+
4492:2
4488:p
4484:=
4479:2
4475:d
4471:+
4466:2
4462:c
4458:+
4453:2
4449:b
4445:+
4440:2
4436:a
4400:.
4393:c
4390:b
4387:+
4384:d
4381:a
4376:)
4372:C
4362:+
4358:A
4348:(
4345:d
4342:c
4339:b
4336:a
4333:2
4327:)
4324:d
4321:b
4318:+
4315:c
4312:a
4309:(
4306:)
4303:d
4300:c
4297:+
4294:b
4291:a
4288:(
4281:=
4278:q
4248:d
4245:c
4242:+
4239:b
4236:a
4231:)
4227:D
4217:+
4213:B
4203:(
4200:d
4197:c
4194:b
4191:a
4188:2
4182:)
4179:c
4176:b
4173:+
4170:d
4167:a
4164:(
4161:)
4158:d
4155:b
4152:+
4149:c
4146:a
4143:(
4136:=
4133:p
4107:.
4101:C
4091:c
4088:b
4085:2
4077:2
4073:c
4069:+
4064:2
4060:b
4054:=
4048:A
4038:d
4035:a
4032:2
4024:2
4020:d
4016:+
4011:2
4007:a
4001:=
3998:q
3969:D
3959:d
3956:c
3953:2
3945:2
3941:d
3937:+
3932:2
3928:c
3922:=
3916:B
3906:b
3903:a
3900:2
3892:2
3888:b
3884:+
3879:2
3875:a
3869:=
3866:p
3658:.
3654:|
3648:1
3644:y
3638:2
3634:x
3625:2
3621:y
3615:1
3611:x
3606:|
3599:2
3596:1
3590:=
3587:K
3574:)
3571:2
3568:y
3566:,
3564:2
3561:x
3558:(
3548:)
3545:1
3542:y
3540:,
3538:1
3535:x
3532:(
3489:,
3485:|
3480:D
3477:B
3469:C
3466:A
3461:|
3454:2
3451:1
3445:=
3442:K
3428:D
3422:B
3416:C
3410:A
3362:,
3357:]
3352:2
3348:)
3344:q
3338:p
3335:(
3327:2
3323:m
3319:4
3316:[
3310:]
3305:2
3301:m
3297:4
3289:2
3285:)
3281:q
3278:+
3275:p
3272:(
3269:[
3261:4
3258:1
3252:=
3249:K
3223:,
3218:]
3213:2
3209:)
3205:n
3199:m
3196:(
3188:2
3184:p
3180:[
3174:]
3169:2
3165:p
3156:2
3152:)
3148:n
3145:+
3142:m
3139:(
3136:[
3128:2
3125:1
3119:=
3116:K
3093:.
3090:)
3085:2
3081:n
3077:+
3072:2
3068:m
3064:(
3061:2
3058:=
3053:2
3049:q
3045:+
3040:2
3036:p
3024:q
3018:p
3012:n
3006:m
2988:.
2981:2
2977:)
2971:2
2967:n
2958:2
2954:m
2950:(
2942:2
2938:q
2932:2
2928:p
2919:2
2916:1
2910:=
2907:K
2884:,
2879:)
2876:q
2870:n
2867:+
2864:m
2861:(
2858:)
2855:q
2852:+
2849:n
2846:+
2843:m
2840:(
2837:)
2834:p
2828:n
2825:+
2822:m
2819:(
2816:)
2813:p
2810:+
2807:n
2804:+
2801:m
2798:(
2790:2
2787:1
2781:=
2778:K
2764:q
2758:p
2752:n
2746:m
2711:.
2704:2
2699:)
2693:2
2689:d
2680:2
2676:b
2667:2
2663:c
2659:+
2654:2
2650:a
2645:(
2635:2
2631:q
2625:2
2621:p
2617:4
2609:4
2606:1
2600:=
2597:K
2574:,
2569:)
2566:q
2563:p
2557:d
2554:b
2551:+
2548:c
2545:a
2542:(
2539:)
2536:q
2533:p
2530:+
2527:d
2524:b
2521:+
2518:c
2515:a
2512:(
2506:4
2503:1
2494:)
2491:d
2485:s
2482:(
2479:)
2476:c
2470:s
2467:(
2464:)
2461:b
2455:s
2452:(
2449:)
2446:a
2440:s
2437:(
2432:=
2429:K
2415:q
2409:p
2403:s
2394:d
2388:c
2382:b
2376:a
2363:-
2359:+
2354:α
2336:,
2329:2
2325:)
2310:b
2307:a
2304:2
2301:+
2296:2
2292:b
2283:2
2279:a
2270:2
2266:d
2262:+
2257:2
2253:c
2249:(
2241:2
2237:d
2231:2
2227:c
2223:4
2215:4
2212:1
2206:+
2192:b
2189:a
2183:2
2180:1
2174:=
2171:K
2157:b
2151:a
2145:α
2139:d
2133:c
2127:b
2121:a
2108:φ
2102:x
2069:)
2062:2
2058:x
2054:2
2048:)
2043:2
2039:d
2035:+
2030:2
2026:b
2022:(
2017:(
2010:)
2003:2
1999:x
1995:2
1989:)
1984:2
1980:c
1976:+
1971:2
1967:a
1963:(
1958:(
1948:2
1945:1
1939:=
1936:K
1922:d
1916:c
1910:b
1904:a
1887:.
1883:|
1877:2
1873:b
1864:2
1860:a
1855:|
1847:|
1833:|
1826:2
1823:1
1817:=
1814:K
1791:.
1787:|
1781:2
1777:d
1768:2
1764:b
1755:2
1751:c
1747:+
1742:2
1738:a
1733:|
1725:|
1711:|
1704:4
1701:1
1695:=
1692:K
1674:θ
1668:θ
1651:.
1647:A
1634:b
1631:a
1628:=
1625:K
1603:.
1599:A
1589:)
1586:c
1583:b
1580:+
1577:d
1574:a
1571:(
1565:2
1562:1
1556:=
1553:K
1530:.
1526:C
1516:c
1513:b
1507:2
1504:1
1498:+
1494:A
1484:d
1481:a
1475:2
1472:1
1466:=
1463:K
1449:d
1443:a
1437:A
1431:c
1425:b
1419:C
1408:C
1404:A
1393:C
1387:A
1381:s
1375:d
1369:c
1363:b
1357:a
1333:)
1330:C
1327:+
1324:A
1321:(
1315:2
1312:1
1301:2
1292:d
1289:c
1286:b
1283:a
1277:)
1274:d
1268:s
1265:(
1262:)
1259:c
1253:s
1250:(
1247:)
1244:b
1238:s
1235:(
1232:)
1229:a
1223:s
1220:(
1215:=
1203:]
1200:)
1197:C
1194:+
1191:A
1188:(
1179:+
1176:1
1173:[
1169:d
1166:c
1163:b
1160:a
1154:2
1151:1
1142:)
1139:d
1133:s
1130:(
1127:)
1124:c
1118:s
1115:(
1112:)
1109:b
1103:s
1100:(
1097:)
1094:a
1088:s
1085:(
1080:=
1073:K
1049:φ
1043:n
1037:m
1019:,
1007:n
1004:m
1001:=
998:K
977:θ
959:2
955:q
952:p
945:=
942:K
931:θ
925:q
919:p
901:,
889:q
886:p
880:2
877:1
871:=
868:K
842:d
834:c
826:b
818:a
808:K
671:.
611:.
592:.
554:.
423:.
379:n
375:S
371:n
354:.
341:=
338:D
332:+
329:C
323:+
320:B
314:+
311:A
247:D
244:C
241:B
238:A
215:D
195:C
175:B
155:A
98:)
94:(
62:4
30:.
23:.
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