4800:
629:
4961:
5985:
20:
7844:
452:
786:
2010:
7765:
7440:
5238:
has three diagonals that meet at a point. From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of tangency; all six of the sides of this hexagon lie on lines tangent to the inscribed circle, so its diagonals meet at a point.
5926:
is made in the form of a tangential quadrilateral, then it will remain tangential no matter how the linkage is flexed, provided the quadrilateral remains convex. (Thus, for example, if a square is deformed into a rhombus it remains tangential, though to a smaller incircle). If one side is held in a
6577:
In 1996, Vaynshtejn was probably the first to prove another beautiful characterization of tangential quadrilaterals, that has later appeared in several magazines and websites. It states that when a convex quadrilateral is divided into four nonoverlapping triangles by its two diagonals, then the
623:
5239:
But two of these diagonals are the same as the diagonals of the tangential quadrilateral, and the third diagonal of the hexagon is the line through two opposite points of tangency. Repeating this same argument with the other two points of tangency completes the proof of the result.
6923:
5433:
6582:. A related result is that the incircles can be exchanged for the excircles to the same triangles (tangent to the sides of the quadrilateral and the extensions of its diagonals). Thus a convex quadrilateral is tangential if and only if the excenters in these four
2884:
2505:
5918:
respectively. Combining the first equality with a previous property, the "vertex centroid" of the tangential quadrilateral coincides with the incenter if and only if the incenter is the midpoint of the line segment connecting the midpoints of the diagonals.
1463:
3852:
6775:
6445:
6299:
6164:
5881:
4325:
4140:
4025:
4473:
7451:
7129:
3653:
3513:
3373:
3233:
3039:
1726:
1027:
5646:
2280:
1625:
1190:
491:
5539:
2143:
7118:
6794:
1812:
4602:
2594:
8038:
5291:
793:(in green) joining the four contact points between the incircle and the sides. Also shown are the tangency chords joining opposite contact points (in red) and the tangent lengths on the sides
287:
632:
A tangential quadrilateral (in blue) with its incircle (dashed line) and the four externally tangent circles (in red), each tangent to a given side and the extensions of the adjacent sides.
1277:
2691:
2315:
446:
913:
384:
6572:
2679:
8215:
5720:
8148:
7959:
713:
1889:
6169:
This characterization had already been proved five years earlier by
Vaynshtejn. In the solution to his problem, a similar characterization was given by Vasilyev and Senderov. If
192:
meet at the center of the incircle. Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter.
4785:
4746:
5966:
1517:
1357:
1321:
1996:
8057:
A tangential quadrilateral is bicentric if and only if its inradius is greater than that of any other tangential quadrilateral having the same sequence of side lengths.
3675:
1935:
7760:{\displaystyle {\frac {(a+p_{1}-q_{1})(c+p_{2}-q_{2})}{(a-p_{1}+q_{1})(c-p_{2}+q_{2})}}={\frac {(b+p_{2}-q_{1})(d+p_{1}-q_{2})}{(b-p_{2}+q_{1})(d-p_{1}+q_{2})}}.}
6683:
6353:
6207:
6072:
5765:
7435:{\displaystyle {\frac {(p_{1}+q_{1}-a)(p_{2}+q_{2}-c)}{(p_{1}+q_{1}+a)(p_{2}+q_{2}+c)}}={\frac {(p_{2}+q_{1}-b)(p_{1}+q_{2}-d)}{(p_{2}+q_{1}+b)(p_{1}+q_{2}+d)}}}
4180:
4031:
3916:
4331:
3519:
3379:
3239:
3099:
6305:
2020:
The four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four
2892:
1643:
940:
6201:
in the same four triangles (from the diagonal intersection to the sides of the quadrilateral), then the quadrilateral is tangential if and only if
5558:
2181:
1540:
618:{\displaystyle \tan {\frac {\angle ABD}{2}}\cdot \tan {\frac {\angle BDC}{2}}=\tan {\frac {\angle ADB}{2}}\cdot \tan {\frac {\angle DBC}{2}}.}
1085:
5452:
2073:
7005:
6347:
are each tangent to one side of the quadrilateral and the extensions of its diagonals). A quadrilateral is tangential if and only if
851:
in the figure) of a tangential quadrilateral are the line segments that connect contact points on opposite sides. These are also the
6918:{\displaystyle {\frac {a}{\triangle (APB)}}+{\frac {c}{\triangle (CPD)}}={\frac {b}{\triangle (BPC)}}+{\frac {d}{\triangle (DPA)}}}
5660:
5101:
1821:
is either of the angles between the diagonals. This formula cannot be used when the tangential quadrilateral is a kite, since then
5428:{\displaystyle {\frac {AB}{CD}}={\frac {IA\cdot IB}{IC\cdot ID}},\quad \quad {\frac {BC}{DA}}={\frac {IB\cdot IC}{ID\cdot IA}}.}
1737:
5281:
The ratio of two opposite sides in a tangential quadrilateral can be expressed in terms of the distances between the incenter
4540:
2513:
7965:
6578:
incenters of the four triangles are concyclic if and only if the quadrilateral is tangential. In fact, the incenters form an
120:
can have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square
6583:
6344:
59:
4803:
Construction of the Newton line (in red) of a tangential quadrilateral (in blue), showing the alignment of the incenter
9161:
129:
212:
8758:
2879:{\displaystyle r={\frac {G+{\sqrt {G^{2}-4r_{1}r_{2}r_{3}r_{4}(r_{1}r_{3}+r_{2}r_{4})}}}{2(r_{1}r_{3}+r_{2}r_{4})}}}
2500:{\displaystyle r=2{\sqrt {\frac {(\sigma -uvx)(\sigma -vxy)(\sigma -xyu)(\sigma -yuv)}{uvxy(uv+xy)(ux+vy)(uy+vx)}}}}
70:. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called
9691:
1209:
8928:
199:, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the
8940:
Chao, Wu Wei; Simeonov, Plamen (2000), "When quadrilaterals have inscribed circles (solution to problem 10698)",
1290:
if and only if the tangential quadrilateral is also cyclic and hence bicentric, this shows that the maximal area
396:
881:
337:
6579:
6508:
8942:
8889:
8589:
2618:
8167:
5672:
8871:
8615:
8103:
7914:
658:
1855:
9284:
9264:
8231:
8051:
153:
1458:{\displaystyle \displaystyle K={\sqrt {abcd}}\sin {\frac {A+C}{2}}={\sqrt {abcd}}\sin {\frac {B+D}{2}}.}
9259:
9216:
9191:
4751:
4712:
5930:
1483:
6999:. Then the quadrilateral is tangential if and only if any one of the following equalities are true:
3898:
respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral
3085:
respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral
1293:
8318:
7825:
5988:
Chao and
Simeonov's characterization in terms of the radii of circles within each of four triangles
4655:
3847:{\displaystyle \sin {\varphi }={\sqrt {\frac {(e+f+g+h)(efg+fgh+ghe+hef)}{(e+f)(f+g)(g+h)(h+e)}}}.}
1957:
1965:
1637:
In fact, the area can be expressed in terms of just two adjacent sides and two opposite angles as
485:
is due to
Iosifescu. He proved in 1954 that a convex quadrilateral has an incircle if and only if
9244:
7896:
4620:
2165:
1895:
1477:
169:
8687:
7770:
5927:
fixed position, then as the quadrilateral is flexed, the incenter traces out a circle of radius
9269:
9154:
8357:"Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral"
5231:
109:. Due to the risk of confusion with a quadrilateral that has a circumcircle, which is called a
1911:
9670:
9610:
9249:
9126:
833:
6770:{\displaystyle {\frac {1}{R_{a}}}+{\frac {1}{R_{c}}}={\frac {1}{R_{b}}}+{\frac {1}{R_{d}}}.}
6440:{\displaystyle {\frac {1}{r_{a}}}+{\frac {1}{r_{c}}}={\frac {1}{r_{b}}}+{\frac {1}{r_{d}}}.}
6294:{\displaystyle {\frac {1}{h_{1}}}+{\frac {1}{h_{3}}}={\frac {1}{h_{2}}}+{\frac {1}{h_{4}}}.}
6159:{\displaystyle {\frac {1}{r_{1}}}+{\frac {1}{r_{3}}}={\frac {1}{r_{2}}}+{\frac {1}{r_{4}}}.}
5876:{\displaystyle {\frac {IM_{p}}{IM_{q}}}={\frac {IA\cdot IC}{IB\cdot ID}}={\frac {e+g}{f+h}}}
9554:
9324:
9254:
9196:
9043:
8633:
8226:
7892:
6587:
4320:{\displaystyle \displaystyle k={\frac {2(efg+fgh+ghe+hef)}{\sqrt {(e+f)(g+h)(e+g)(f+h)}}},}
4135:{\displaystyle \displaystyle q={\sqrt {{\frac {f+h}{e+g}}{\Big (}(e+g)(f+h)+4eg{\Big )}}}.}
4020:{\displaystyle \displaystyle p={\sqrt {{\frac {e+g}{f+h}}{\Big (}(e+g)(f+h)+4fh{\Big )}}},}
2161:
1520:
1473:
1195:
Furthermore, the area of a tangential quadrilateral can be expressed in terms of the sides
177:
165:
110:
9041:
De
Villiers, Michael (2011), "Equiangular cyclic and equilateral circumscribed polygons",
4468:{\displaystyle \displaystyle l={\frac {2(efg+fgh+ghe+hef)}{\sqrt {(e+h)(f+g)(e+g)(f+h)}}}}
8:
9660:
9635:
9605:
9600:
9559:
9274:
8679:
8236:
6198:
5167:
801:. These four points define a new quadrilateral inside of the initial quadrilateral: the
9665:
9206:
9091:
9064:
9015:
8988:
8906:
8830:
8776:
8732:
8705:
8654:
8440:
8356:
8287:
8241:
5049:
4799:
3648:{\displaystyle \sin {\frac {D}{2}}={\sqrt {\frac {efg+fgh+ghe+hef}{(h+e)(h+f)(h+g)}}}.}
3508:{\displaystyle \sin {\frac {C}{2}}={\sqrt {\frac {efg+fgh+ghe+hef}{(g+e)(g+f)(g+h)}}},}
3368:{\displaystyle \sin {\frac {B}{2}}={\sqrt {\frac {efg+fgh+ghe+hef}{(f+e)(f+g)(f+h)}}},}
3228:{\displaystyle \sin {\frac {A}{2}}={\sqrt {\frac {efg+fgh+ghe+hef}{(e+f)(e+g)(e+h)}}},}
2156:
is its semiperimeter. For a tangential quadrilateral with given sides, the inradius is
828:
in the figure to the right) of a tangential quadrilateral are the line segments from a
83:
28:
8803:
8496:
8413:
9645:
9239:
9147:
9122:
8575:
2157:
1945:
1469:
829:
8552:
4964:
A tangential quadrilateral is partitioned in four triangles meeting at its incenter
113:
or inscribed quadrilateral, it is preferable not to use any of the last five names.
9174:
8951:
8898:
8598:
5923:
5227:
628:
8972:
Vaynshtejn, I.; Vasilyev, N.; Senderov, V. (1995), "(Solution to problem) M1495",
9640:
9620:
9615:
9585:
9304:
9279:
9211:
7793:
4933:, and the extensions of opposite sides in its contact quadrilateral intersect at
4627:
3034:{\displaystyle G=r_{1}r_{2}r_{3}+r_{2}r_{3}r_{4}+r_{3}r_{4}r_{1}+r_{4}r_{1}r_{2}}
141:
8757:
Gutierrez, Antonio, "Circumscribed
Quadrilateral, Diagonal, Chord, Proportion",
8441:"Similar Metric Characterizations of Tangential and Extangential Quadrilaterals"
7814:
The two line segments connecting opposite points of tangency have equal lengths.
6677:, then another condition is that the quadrilateral is tangential if and only if
1721:{\displaystyle K=ab\sin {\frac {B}{2}}\csc {\frac {D}{2}}\sin {\frac {B+D}{2}}.}
9650:
9630:
9595:
9590:
9221:
9201:
6064:
5664:
5242:
If the extensions of opposite sides in a tangential quadrilateral intersect at
4960:
4925:
If the extensions of opposite sides in a tangential quadrilateral intersect at
2285:
The inradius can also be expressed in terms of the distances from the incenter
1022:{\displaystyle \displaystyle K={\tfrac {1}{2}}{\sqrt {p^{2}q^{2}-(ac-bd)^{2}}}}
328:
189:
44:
9685:
9625:
9476:
9369:
9289:
9231:
7834:
The center of the incircle lies on the diagonal that is the axis of symmetry.
7808:
7771:
Conditions for a tangential quadrilateral to be another type of quadrilateral
5984:
5235:
5096:
of the quadrilateral into two equal parts. More importantly, the Nagel point
4612:
1833:
As indirectly noted above, the area of a tangential quadrilateral with sides
923:
200:
196:
47:
19:
6063:
respectively. Chao and
Simeonov proved that the quadrilateral is tangential
5070:
of the tangential quadrilateral is defined as the intersection of the lines
805:
which is cyclic as it is inscribed in the initial quadrilateral's incircle.
9655:
9525:
9481:
9445:
9435:
9430:
6479:
5641:{\displaystyle IA\cdot IC+IB\cdot ID={\sqrt {AB\cdot BC\cdot CD\cdot DA}}.}
4915:
4616:
2275:{\displaystyle \displaystyle r={\sqrt {\frac {efg+fgh+ghe+hef}{e+f+g+h}}}.}
1332:
161:
7843:
6016:, there are the following characterizations of tangential quadrilaterals.
458:
Another necessary and sufficient condition is that a convex quadrilateral
9564:
9471:
9450:
9440:
5275:
5234:, which states that a hexagon all of whose sides are tangent to a single
4919:
1944:
is the inradius. There is equality if and only if the quadrilateral is a
1620:{\displaystyle K=\left(IA\cdot IC+IB\cdot ID\right)\sin {\frac {A+C}{2}}}
770:
respectively and the extensions of the adjacent two sides for each side.
8887:
Barton, Helen (1926), "On a circle attached to a collapsible four-bar",
9569:
9425:
9415:
9299:
8910:
2021:
2001:
with equality if and only if the tangential quadrilateral is a square.
157:
451:
89:
Other less frequently used names for this class of quadrilaterals are
9544:
9534:
9511:
9501:
9491:
9420:
9329:
9294:
9131:
9016:"A new formula concerning the diagonals and sides of a quadrilateral"
8086:
5093:
2036:
1185:{\displaystyle \displaystyle K={\sqrt {(e+f+g+h)(efg+fgh+ghe+hef)}}.}
316:
173:
121:
8955:
8902:
8602:
4968:, their orthocenters (purple) and the intersection of the diagonals
4626:
have equal lengths if and only if the tangential quadrilateral is a
785:
152:. The kites are exactly the tangential quadrilaterals that are also
9549:
9539:
9496:
9455:
9384:
9374:
9364:
9183:
9065:"On a circle containing the incenters of tangential quadrilaterals"
7801:
5740:
5534:{\displaystyle AB\cdot BC=IB^{2}+{\frac {IA\cdot IB\cdot IC}{ID}}.}
4851:
2040:
2009:
1524:
931:
852:
117:
9139:
5052:
of these points with respect to the corresponding sides (that is,
2138:{\displaystyle r={\frac {K}{s}}={\frac {K}{a+c}}={\frac {K}{b+d}}}
2051:
The inradius in a tangential quadrilateral with consecutive sides
9506:
9486:
9399:
9394:
9389:
9379:
9354:
9309:
9170:
9120:
7781:
2028:
1323:
occurs if and only if the tangential quadrilateral is bicentric.
471:
145:
51:
5438:
The product of two adjacent sides in a tangential quadrilateral
462:
is tangential if and only if the incircles in the two triangles
62:
of the quadrilateral or its inscribed circle, its center is the
9314:
8923:
Bogomolny, Alexander, "When A Quadrilateral Is
Inscriptible?",
8858:
103 Trigonometry
Problems From the Training of the USA IMO Team
8497:"A condition for a circumscriptible quadrilateral to be cyclic"
7113:{\displaystyle ap_{2}q_{2}+cp_{1}q_{1}=bp_{1}q_{2}+dp_{2}q_{1}}
6601:, is tangential if and only if the four excenters in triangles
774:
149:
55:
2681:
respectively, then the inradius of a tangential quadrilateral
9359:
8316:
Bryant, Victor; Duncan, John (2010), "Wheels within wheels",
5979:
3090:
754:
are the radii in the circles externally tangent to the sides
132:
a quadrilateral must satisfy to be able to have an incircle.
8587:
Hoyt, John P. (1986), "Maximizing the Area of a
Trapezium",
7899:) if and only if any one of the following conditions hold:
7796:
if and only if any one of the following conditions is true:
777:
are known in the four subtriangles formed by the diagonals.
8682:(2016), An Inradii Relation in Inscriptible Quadrilateral,
2032:
1530:
Another formula for the area of a tangential quadrilateral
869:
477:
A characterization regarding the angles formed by diagonal
8971:
832:
to the points of contact. From each vertex, there are two
1807:{\displaystyle K={\tfrac {1}{2}}|(ac-bd)\tan {\theta }|,}
1059:
The area can also be expressed in terms of just the four
8831:"The two incenters of an arbitrary convex quadrilateral"
8829:
Dergiades, Nikolaos; Christodoulou, Dimitris M. (2017),
6047:
denote the radii of the incircles in the four triangles
4597:{\displaystyle {\frac {k^{2}}{l^{2}}}={\frac {bd}{ac}}.}
2589:{\displaystyle \sigma ={\tfrac {1}{2}}(uvx+vxy+xyu+yuv)}
1901:
According to T. A. Ivanova (in 1976), the semiperimeter
8033:{\displaystyle {\frac {AC}{BD}}={\frac {AW+CY}{BX+DZ}}}
5274:
The incenter of a tangential quadrilateral lies on its
4615:
if and only if the tangential quadrilateral also has a
4170:
of a tangential quadrilateral, then the lengths of the
8828:
4832:(in green) joining the intersection of opposing sides.
4756:
4717:
2524:
1748:
952:
636:
Further, a convex quadrilateral with successive sides
8288:"More Characterizations of Tangential Quadrilaterals"
8170:
8161:
are the parallel sides of a trapezoid if and only if
8106:
7968:
7917:
7454:
7132:
7008:
6797:
6686:
6511:
6356:
6210:
6075:
5933:
5768:
5675:
5561:
5455:
5294:
5221:
4754:
4715:
4543:
4534:. The squared ratio of the tangency chords satisfies
4335:
4334:
4184:
4183:
4035:
4034:
3920:
3919:
3678:
3522:
3382:
3242:
3102:
2895:
2694:
2621:
2516:
2318:
2185:
2184:
2076:
1968:
1914:
1858:
1740:
1646:
1543:
1486:
1361:
1360:
1296:
1212:
1089:
1088:
944:
943:
885:
884:
780:
661:
494:
400:
399:
341:
340:
215:
8804:"On Two Remarkable Lines Related to a Quadrilateral"
7851:: the contact quadrilateral (pink) is orthodiagonal.
58:
within the quadrilateral. This circle is called the
8989:"Characterizations of Orthodiagonal Quadrilaterals"
6939:Denote the segments that the diagonal intersection
2027:If a line cuts a tangential quadrilateral into two
8209:
8142:
8032:
7953:
7759:
7434:
7112:
6917:
6769:
6566:
6439:
6293:
6158:
6008:formed by the diagonals in a convex quadrilateral
5960:
5875:
5714:
5640:
5533:
5427:
5226:The two diagonals and the two tangency chords are
4779:
4740:
4596:
4467:
4319:
4134:
4019:
3846:
3647:
3507:
3367:
3227:
3033:
2878:
2673:
2588:
2499:
2274:
2137:
1990:
1929:
1883:
1806:
1720:
1619:
1511:
1457:
1315:
1271:
1184:
1021:
907:
707:
617:
440:
378:
281:
82:. Tangential quadrilaterals are a special case of
8414:"Characterizations of a Tangential Quadrilateral"
5278:(which connects the midpoints of the diagonals).
4121:
4072:
4006:
3957:
1079:, then the tangential quadrilateral has the area
9683:
2289:to the vertices of the tangential quadrilateral
1825:is 90° and the tangent function is not defined.
8777:"Characterizations of Bicentric Quadrilaterals"
8655:"On the inradius of a tangential quadrilateral"
8390:
5230:. One way to see this is as a limiting case of
1032:which gives the area in terms of the diagonals
282:{\displaystyle a+c=b+d={\frac {a+b+c+d}{2}}=s.}
8925:Interactive Mathematics Miscellany and Puzzles
8081:respectively, then a tangential quadrilateral
7887:respectively, then a tangential quadrilateral
7784:if and only if its opposite angles are equal.
6304:Another similar characterization concerns the
5548:is the incenter of a tangential quadrilateral
140:Examples of tangential quadrilaterals are the
9155:
8967:
8965:
8855:
8770:
8768:
8766:
8699:
8697:
8695:
8521:
8281:
8279:
8277:
4870:, and if the pairs of opposite sides meet at
1272:{\displaystyle K={\sqrt {abcd-(eg-fh)^{2}}}.}
863:
8939:
8706:"When is a Tangential Quadrilateral a Kite?"
8473:
8315:
8275:
8273:
8271:
8269:
8267:
8265:
8263:
8261:
8259:
8257:
3093:of the quadrilateral can be calculated from
797:The incircle is tangent to each side at one
789:A tagential quadrilateral (in blue) and its
450:
311:If opposite sides in a convex quadrilateral
164:. If a quadrilateral is both tangential and
23:A tangential quadrilateral with its incircle
9040:
8548:
8546:
8544:
8542:
8540:
8538:
8536:
8534:
8407:
8405:
8403:
8386:
8384:
8382:
8380:
8378:
8350:
8348:
8346:
8344:
8342:
8340:
8338:
8336:
8334:
5751:respectively in a tangential quadrilateral
4862:respectively in a tangential quadrilateral
1335:formula for the area in terms of the sides
292:Conversely a convex quadrilateral in which
16:Polygon whose four sides all touch a circle
9162:
9148:
8962:
8763:
8692:
7838:
7818:
5980:Characterizations in the four subtriangles
4666:is shorter than the one between the sides
4171:
4167:
3879:
3659:
3066:
2172:
1523:. This can be proved in another way using
1060:
875:of a tangential quadrilateral is given by
441:{\displaystyle \displaystyle AE-EC=AF-FC:}
9089:
8986:
8801:
8795:
8774:
8730:
8703:
8652:
8438:
8411:
8354:
8285:
8254:
7831:The products of opposite sides are equal.
4646:is longer than the one between the sides
1326:
908:{\displaystyle \displaystyle K=r\cdot s,}
379:{\displaystyle \displaystyle BE+BF=DE+DF}
8631:Hoyt, John P. (1984), "Quickies, Q694",
8531:
8469:
8467:
8465:
8463:
8461:
8400:
8391:Andreescu, Titu; Enescu, Bogdan (2006),
8375:
8331:
8311:
8309:
8065:If the incircle is tangent to the sides
8043:The first of these three means that the
7855:If the incircle is tangent to the sides
7842:
7800:The area is one half the product of the
6567:{\displaystyle R_{1}+R_{3}=R_{2}+R_{4}.}
5983:
4976:If the incircle is tangent to the sides
4959:
4798:
2008:
1905:of a tangential quadrilateral satisfies
784:
627:
188:In a tangential quadrilateral, the four
18:
9092:"The diagonal point triangle revisited"
8733:"The Area of a Bicentric Quadrilateral"
8490:
8488:
8486:
8060:
2674:{\displaystyle r_{1},r_{2},r_{3},r_{4}}
2013:Tangential quadrilateral with inradius
2004:
9684:
8886:
8856:Andreescu, Titu; Feng, Zuming (2005),
8566:
8564:
8555:Circumscribed quadrilaterals revisited
8210:{\displaystyle AW\cdot BW=CY\cdot DY.}
5715:{\displaystyle IA\cdot IC=IB\cdot ID.}
481:and the four sides of a quadrilateral
125:
9143:
9121:
9013:
8494:
8458:
8306:
8143:{\displaystyle AW\cdot DY=BW\cdot CY}
7954:{\displaystyle AW\cdot CY=BW\cdot DY}
6498:respectively, then the quadrilateral
6343:in the same four triangles (the four
5258:is perpendicular to the extension of
5134:and where the diagonals intersect at
4634:The tangency chord between the sides
2152:is the area of the quadrilateral and
1894:with equality if and only if it is a
1534:that involves two opposite angles is
708:{\displaystyle R_{a}R_{c}=R_{b}R_{d}}
172:, and if it is both tangential and a
9062:
9056:
8630:
8586:
8526:, Cambridge Univ. Press, p. 203
8522:Siddons, A.W.; Hughes, R.T. (1929),
8483:
4709:, then the ratio of tangent lengths
3902:, then the lengths of the diagonals
2039:, then that line passes through the
1884:{\displaystyle K\leq {\sqrt {abcd}}}
1468:For given side lengths, the area is
183:
9169:
8561:
6669:respectively opposite the vertices
6012:, where the diagonals intersect at
4794:
4478:where the tangency chord of length
1199:and the successive tangent lengths
130:necessary and sufficient conditions
13:
9083:
6894:
6864:
6834:
6804:
6580:orthodiagonal cyclic quadrilateral
5222:Concurrent and perpendicular lines
4918:. The line containing them is the
4145:
781:Contact points and tangent lengths
594:
564:
534:
504:
14:
9703:
9114:
8872:"Determine ratio OM/ON", Post at
8474:Durell, C.V.; Robson, A. (2003),
6653:are the exradii in the triangles
6597:, with diagonals intersecting at
5250:, and the diagonals intersect at
5111:are collinear in this order, and
5092:. Both of these lines divide the
4807:, the midpoints of the diagonals
4780:{\displaystyle {\tfrac {BM}{DM}}}
4741:{\displaystyle {\tfrac {BW}{DY}}}
3044:
1056:of the tangential quadrilateral.
7792:A tangential quadrilateral is a
7780:A tangential quadrilateral is a
6780:Further, a convex quadrilateral
5992:In the nonoverlapping triangles
5961:{\displaystyle {\sqrt {abcd}}/s}
1512:{\displaystyle K={\sqrt {abcd}}}
135:
9034:
9007:
8980:
8933:
8917:
8880:
8865:
8849:
8822:
8751:
8724:
8673:
8646:
8624:
8609:
8580:
8478:, Dover reprint, pp. 28–30
8393:Mathematical Olympiad Treasures
6784:with diagonals intersecting at
5360:
5359:
5123:of a tangential quadrilateral.
2160:when the quadrilateral is also
1948:. This means that for the area
1828:
1472:when the quadrilateral is also
80:circumscriptible quadrilaterals
8860:, Birkhäuser, pp. 176–177
8515:
8432:
7748:
7716:
7713:
7681:
7676:
7644:
7641:
7609:
7597:
7565:
7562:
7530:
7525:
7493:
7490:
7458:
7426:
7394:
7391:
7359:
7354:
7322:
7319:
7287:
7275:
7243:
7240:
7208:
7203:
7171:
7168:
7136:
6909:
6897:
6879:
6867:
6849:
6837:
6819:
6807:
5972:are the sides in sequence and
5655:in a tangential quadrilateral
5285:and the vertices according to
5126:In a tangential quadrilateral
4642:in a tangential quadrilateral
4510:connects the sides of lengths
4482:connects the sides of lengths
4458:
4446:
4443:
4431:
4428:
4416:
4413:
4401:
4396:
4348:
4307:
4295:
4292:
4280:
4277:
4265:
4262:
4250:
4245:
4197:
4104:
4092:
4089:
4077:
3989:
3977:
3974:
3962:
3834:
3822:
3819:
3807:
3804:
3792:
3789:
3777:
3772:
3724:
3721:
3697:
3635:
3623:
3620:
3608:
3605:
3593:
3495:
3483:
3480:
3468:
3465:
3453:
3355:
3343:
3340:
3328:
3325:
3313:
3215:
3203:
3200:
3188:
3185:
3173:
2870:
2824:
2814:
2768:
2599:If the incircles in triangles
2583:
2535:
2490:
2472:
2469:
2451:
2448:
2430:
2413:
2395:
2392:
2374:
2371:
2353:
2350:
2332:
1797:
1782:
1764:
1760:
1731:Still another area formula is
1316:{\displaystyle {\sqrt {abcd}}}
1255:
1236:
1173:
1125:
1122:
1098:
1007:
988:
855:of the contact quadrilateral.
72:circumscribable quadrilaterals
1:
8943:American Mathematical Monthly
8890:American Mathematical Monthly
8590:American Mathematical Monthly
8247:
6788:is tangential if and only if
6502:is tangential if and only if
4972:(in green) are all colinear,.
652:is tangential if and only if
76:circumscribing quadrilaterals
66:and its radius is called the
8395:, Birkhäuser, pp. 64–68
4787:of the segments of diagonal
4677:If tangential quadrilateral
3857:
1991:{\displaystyle K\geq 4r^{2}}
148:, which in turn include the
7:
8412:Minculete, Nicusor (2009),
8232:Ex-tangential quadrilateral
8220:
8052:orthodiagonal quadrilateral
5902:are the tangent lengths at
5269:
2046:
1351:and two opposite angles is
41:circumscribed quadrilateral
10:
9708:
9127:"Tangential Quadrilateral"
9090:Josefsson, Martin (2014),
8987:Josefsson, Martin (2012),
8802:Myakishev, Alexei (2006),
8775:Josefsson, Martin (2010),
8731:Josefsson, Martin (2011),
8704:Josefsson, Martin (2011),
8653:Josefsson, Martin (2010),
8439:Josefsson, Martin (2012),
8355:Josefsson, Martin (2010),
8286:Josefsson, Martin (2011),
7847:A bicentric quadrilateral
7775:
6932:) is the area of triangle
5119:. This line is called the
2175:, the incircle has radius
1519:since opposite angles are
864:Non-trigonometric formulas
103:circumcyclic quadrilateral
95:inscriptible quadrilateral
91:inscriptable quadrilateral
9578:
9524:
9464:
9408:
9347:
9338:
9230:
9182:
99:inscribable quadrilateral
8495:Hajja, Mowaffaq (2008),
8319:The Mathematical Gazette
6478:denote the radii in the
4697:, and if tangency chord
4607:The two tangency chords
4506:, and the one of length
1930:{\displaystyle s\geq 4r}
327:, then it is tangential
33:tangential quadrilateral
9692:Types of quadrilaterals
9063:Hess, Albrecht (2014),
7839:Bicentric quadrilateral
7787:
6593:A convex quadrilateral
4941:, then the four points
2166:bicentric quadrilateral
1896:bicentric quadrilateral
1478:bicentric quadrilateral
858:
170:bicentric quadrilateral
107:co-cyclic quadrilateral
50:whose sides all can be
8874:Art of Problem Solving
8760:, Accessed 2012-04-09.
8618:Art of Problem Solving
8211:
8144:
8034:
7955:
7852:
7761:
7436:
7114:
6919:
6771:
6617:opposite the vertices
6586:are the vertices of a
6568:
6441:
6295:
6160:
5989:
5976:is the semiperimeter.
5962:
5877:
5716:
5642:
5535:
5429:
4973:
4922:of the quadrilateral.
4885:being the midpoint of
4833:
4781:
4742:
4598:
4469:
4321:
4136:
4021:
3848:
3658:The angle between the
3649:
3509:
3369:
3229:
3035:
2880:
2675:
2590:
2501:
2276:
2139:
2017:
1992:
1931:
1885:
1808:
1722:
1621:
1513:
1459:
1327:Trigonometric formulas
1317:
1273:
1186:
1023:
909:
803:contact quadrilateral,
794:
775:more characterizations
709:
633:
619:
455:
442:
380:
283:
206:of the quadrilateral:
24:
9014:Hoehn, Larry (2011),
8476:Advanced Trigonometry
8212:
8145:
8045:contact quadrilateral
8035:
7956:
7846:
7817:One pair of opposite
7762:
7437:
7115:
6920:
6772:
6569:
6442:
6296:
6161:
5987:
5963:
5878:
5717:
5663:of the quadrilateral
5643:
5536:
5430:
5066:and so on), then the
5020:respectively, and if
4963:
4802:
4782:
4743:
4599:
4470:
4322:
4137:
4022:
3849:
3650:
3510:
3370:
3230:
3036:
2881:
2676:
2591:
2502:
2277:
2140:
2012:
1993:
1932:
1886:
1809:
1723:
1622:
1514:
1460:
1318:
1274:
1187:
1024:
934:. Another formula is
910:
791:contact quadrilateral
788:
710:
631:
620:
454:
443:
381:
284:
37:tangent quadrilateral
22:
9395:Nonagon/Enneagon (9)
9325:Tangential trapezoid
9044:Mathematical Gazette
8680:Bogomolny, Alexander
8634:Mathematics Magazine
8577:, 1998, pp. 156–157.
8227:Circumscribed circle
8168:
8104:
8089:with parallel sides
8061:Tangential trapezoid
7966:
7915:
7906:is perpendicular to
7452:
7130:
7006:
6795:
6684:
6588:cyclic quadrilateral
6509:
6354:
6208:
6073:
5931:
5766:
5673:
5559:
5453:
5292:
4752:
4713:
4701:intersects diagonal
4681:has tangency points
4541:
4332:
4181:
4032:
3917:
3676:
3520:
3380:
3240:
3100:
2893:
2692:
2619:
2514:
2316:
2182:
2074:
2005:Partition properties
1966:
1912:
1856:
1738:
1644:
1541:
1521:supplementary angles
1484:
1358:
1294:
1210:
1086:
941:
882:
659:
492:
397:
338:
308:must be tangential.
213:
178:tangential trapezoid
144:, which include the
111:cyclic quadrilateral
9507:Megagon (1,000,000)
9275:Isosceles trapezoid
9099:Forum Geometricorum
9072:Forum Geometricorum
9023:Forum Geometricorum
8996:Forum Geometricorum
8838:Forum Geometricorum
8811:Forum Geometricorum
8784:Forum Geometricorum
8740:Forum Geometricorum
8713:Forum Geometricorum
8662:Forum Geometricorum
8504:Forum Geometricorum
8448:Forum Geometricorum
8421:Forum Geometricorum
8364:Forum Geometricorum
8326:(November): 502–505
8295:Forum Geometricorum
8237:Tangential triangle
7828:have equal lengths.
7821:have equal lengths.
5659:coincides with the
5232:Brianchon's theorem
5107:, and the incenter
5050:isotomic conjugates
4654:if and only if the
84:tangential polygons
9477:Icositetragon (24)
9123:Weisstein, Eric W.
8572:Euclidean Geometry
8242:Tangential polygon
8207:
8140:
8030:
7951:
7853:
7807:The diagonals are
7757:
7432:
7110:
6915:
6767:
6625:are concyclic. If
6564:
6437:
6291:
6156:
5990:
5958:
5873:
5712:
5638:
5531:
5425:
5186:. Then the points
4974:
4889:, then the points
4834:
4777:
4775:
4738:
4736:
4658:between the sides
4594:
4465:
4464:
4317:
4316:
4132:
4131:
4017:
4016:
3844:
3645:
3505:
3365:
3225:
3069:from the vertices
3031:
2876:
2671:
2586:
2533:
2497:
2272:
2271:
2135:
2018:
1988:
1927:
1881:
1804:
1757:
1718:
1617:
1509:
1455:
1454:
1313:
1269:
1182:
1181:
1019:
1018:
961:
905:
904:
795:
705:
634:
615:
456:
438:
437:
376:
375:
279:
128:below states what
29:Euclidean geometry
25:
9679:
9678:
9520:
9519:
9497:Myriagon (10,000)
9482:Triacontagon (30)
9446:Heptadecagon (17)
9436:Pentadecagon (15)
9431:Tetradecagon (14)
9370:Quadrilateral (4)
9240:Antiparallelogram
8553:Grinberg, Darij,
8028:
7987:
7752:
7601:
7430:
7279:
6973:divides diagonal
6943:divides diagonal
6913:
6883:
6853:
6823:
6762:
6742:
6722:
6702:
6432:
6412:
6392:
6372:
6286:
6266:
6246:
6226:
6151:
6131:
6111:
6091:
5948:
5871:
5842:
5801:
5743:of the diagonals
5661:"vertex centroid"
5633:
5526:
5420:
5379:
5354:
5313:
5266:is the incenter.
4854:of the diagonals
4774:
4748:equals the ratio
4735:
4589:
4566:
4462:
4461:
4311:
4310:
4126:
4068:
4011:
3953:
3839:
3838:
3640:
3639:
3537:
3500:
3499:
3397:
3360:
3359:
3257:
3220:
3219:
3117:
2874:
2817:
2532:
2495:
2494:
2266:
2265:
2133:
2112:
2091:
1879:
1756:
1713:
1689:
1673:
1634:is the incenter.
1615:
1507:
1449:
1425:
1406:
1382:
1311:
1264:
1176:
1016:
960:
836:tangent lengths.
610:
580:
550:
520:
268:
195:According to the
184:Characterizations
176:, it is called a
168:, it is called a
160:is a kite with a
126:characterizations
9699:
9492:Chiliagon (1000)
9472:Icositrigon (23)
9451:Octadecagon (18)
9441:Hexadecagon (16)
9345:
9344:
9164:
9157:
9150:
9141:
9140:
9136:
9135:
9108:
9106:
9096:
9087:
9081:
9079:
9069:
9060:
9054:
9052:
9051:(March): 102–107
9038:
9032:
9030:
9020:
9011:
9005:
9003:
8993:
8984:
8978:
8977:
8969:
8960:
8958:
8937:
8931:
8921:
8915:
8913:
8884:
8878:
8869:
8863:
8861:
8853:
8847:
8845:
8835:
8826:
8820:
8818:
8808:
8799:
8793:
8791:
8781:
8772:
8761:
8755:
8749:
8747:
8737:
8728:
8722:
8720:
8710:
8701:
8690:
8677:
8671:
8669:
8659:
8650:
8644:
8642:
8628:
8622:
8613:
8607:
8605:
8584:
8578:
8568:
8559:
8550:
8529:
8527:
8519:
8513:
8511:
8501:
8492:
8481:
8479:
8471:
8456:
8455:
8445:
8436:
8430:
8428:
8418:
8409:
8398:
8396:
8388:
8373:
8371:
8361:
8352:
8329:
8327:
8313:
8304:
8302:
8292:
8283:
8216:
8214:
8213:
8208:
8149:
8147:
8146:
8141:
8039:
8037:
8036:
8031:
8029:
8027:
8010:
7993:
7988:
7986:
7978:
7970:
7960:
7958:
7957:
7952:
7766:
7764:
7763:
7758:
7753:
7751:
7747:
7746:
7734:
7733:
7712:
7711:
7699:
7698:
7679:
7675:
7674:
7662:
7661:
7640:
7639:
7627:
7626:
7607:
7602:
7600:
7596:
7595:
7583:
7582:
7561:
7560:
7548:
7547:
7528:
7524:
7523:
7511:
7510:
7489:
7488:
7476:
7475:
7456:
7441:
7439:
7438:
7433:
7431:
7429:
7419:
7418:
7406:
7405:
7384:
7383:
7371:
7370:
7357:
7347:
7346:
7334:
7333:
7312:
7311:
7299:
7298:
7285:
7280:
7278:
7268:
7267:
7255:
7254:
7233:
7232:
7220:
7219:
7206:
7196:
7195:
7183:
7182:
7161:
7160:
7148:
7147:
7134:
7119:
7117:
7116:
7111:
7109:
7108:
7099:
7098:
7083:
7082:
7073:
7072:
7057:
7056:
7047:
7046:
7031:
7030:
7021:
7020:
6969:, and similarly
6924:
6922:
6921:
6916:
6914:
6912:
6889:
6884:
6882:
6859:
6854:
6852:
6829:
6824:
6822:
6799:
6776:
6774:
6773:
6768:
6763:
6761:
6760:
6748:
6743:
6741:
6740:
6728:
6723:
6721:
6720:
6708:
6703:
6701:
6700:
6688:
6573:
6571:
6570:
6565:
6560:
6559:
6547:
6546:
6534:
6533:
6521:
6520:
6446:
6444:
6443:
6438:
6433:
6431:
6430:
6418:
6413:
6411:
6410:
6398:
6393:
6391:
6390:
6378:
6373:
6371:
6370:
6358:
6300:
6298:
6297:
6292:
6287:
6285:
6284:
6272:
6267:
6265:
6264:
6252:
6247:
6245:
6244:
6232:
6227:
6225:
6224:
6212:
6165:
6163:
6162:
6157:
6152:
6150:
6149:
6137:
6132:
6130:
6129:
6117:
6112:
6110:
6109:
6097:
6092:
6090:
6089:
6077:
5967:
5965:
5964:
5959:
5954:
5949:
5935:
5924:four-bar linkage
5882:
5880:
5879:
5874:
5872:
5870:
5859:
5848:
5843:
5841:
5824:
5807:
5802:
5800:
5799:
5798:
5785:
5784:
5783:
5770:
5721:
5719:
5718:
5713:
5647:
5645:
5644:
5639:
5634:
5599:
5540:
5538:
5537:
5532:
5527:
5525:
5517:
5491:
5486:
5485:
5434:
5432:
5431:
5426:
5421:
5419:
5402:
5385:
5380:
5378:
5370:
5362:
5355:
5353:
5336:
5319:
5314:
5312:
5304:
5296:
4795:Collinear points
4786:
4784:
4783:
4778:
4776:
4773:
4765:
4757:
4747:
4745:
4744:
4739:
4737:
4734:
4726:
4718:
4603:
4601:
4600:
4595:
4590:
4588:
4580:
4572:
4567:
4565:
4564:
4555:
4554:
4545:
4474:
4472:
4471:
4466:
4463:
4400:
4399:
4343:
4326:
4324:
4323:
4318:
4312:
4249:
4248:
4192:
4141:
4139:
4138:
4133:
4127:
4125:
4124:
4076:
4075:
4069:
4067:
4056:
4045:
4043:
4026:
4024:
4023:
4018:
4012:
4010:
4009:
3961:
3960:
3954:
3952:
3941:
3930:
3928:
3853:
3851:
3850:
3845:
3840:
3837:
3775:
3695:
3694:
3689:
3654:
3652:
3651:
3646:
3641:
3638:
3591:
3544:
3543:
3538:
3530:
3514:
3512:
3511:
3506:
3501:
3498:
3451:
3404:
3403:
3398:
3390:
3374:
3372:
3371:
3366:
3361:
3358:
3311:
3264:
3263:
3258:
3250:
3234:
3232:
3231:
3226:
3221:
3218:
3171:
3124:
3123:
3118:
3110:
3040:
3038:
3037:
3032:
3030:
3029:
3020:
3019:
3010:
3009:
2997:
2996:
2987:
2986:
2977:
2976:
2964:
2963:
2954:
2953:
2944:
2943:
2931:
2930:
2921:
2920:
2911:
2910:
2885:
2883:
2882:
2877:
2875:
2873:
2869:
2868:
2859:
2858:
2846:
2845:
2836:
2835:
2819:
2818:
2813:
2812:
2803:
2802:
2790:
2789:
2780:
2779:
2767:
2766:
2757:
2756:
2747:
2746:
2737:
2736:
2721:
2720:
2711:
2702:
2680:
2678:
2677:
2672:
2670:
2669:
2657:
2656:
2644:
2643:
2631:
2630:
2595:
2593:
2592:
2587:
2534:
2525:
2506:
2504:
2503:
2498:
2496:
2493:
2416:
2330:
2329:
2281:
2279:
2278:
2273:
2267:
2264:
2241:
2194:
2193:
2171:In terms of the
2144:
2142:
2141:
2136:
2134:
2132:
2118:
2113:
2111:
2097:
2092:
2084:
1997:
1995:
1994:
1989:
1987:
1986:
1936:
1934:
1933:
1928:
1890:
1888:
1887:
1882:
1880:
1866:
1813:
1811:
1810:
1805:
1800:
1795:
1763:
1758:
1749:
1727:
1725:
1724:
1719:
1714:
1709:
1698:
1690:
1682:
1674:
1666:
1626:
1624:
1623:
1618:
1616:
1611:
1600:
1592:
1588:
1518:
1516:
1515:
1510:
1508:
1494:
1464:
1462:
1461:
1456:
1450:
1445:
1434:
1426:
1412:
1407:
1402:
1391:
1383:
1369:
1322:
1320:
1319:
1314:
1312:
1298:
1278:
1276:
1275:
1270:
1265:
1263:
1262:
1220:
1191:
1189:
1188:
1183:
1177:
1097:
1028:
1026:
1025:
1020:
1017:
1015:
1014:
984:
983:
974:
973:
964:
962:
953:
914:
912:
911:
906:
799:point of contact
714:
712:
711:
706:
704:
703:
694:
693:
681:
680:
671:
670:
624:
622:
621:
616:
611:
606:
592:
581:
576:
562:
551:
546:
532:
521:
516:
502:
447:
445:
444:
439:
385:
383:
382:
377:
288:
286:
285:
280:
269:
264:
241:
35:(sometimes just
9707:
9706:
9702:
9701:
9700:
9698:
9697:
9696:
9682:
9681:
9680:
9675:
9574:
9528:
9516:
9460:
9426:Tridecagon (13)
9416:Hendecagon (11)
9404:
9340:
9334:
9305:Right trapezoid
9226:
9178:
9168:
9117:
9112:
9111:
9094:
9088:
9084:
9067:
9061:
9057:
9039:
9035:
9018:
9012:
9008:
8991:
8985:
8981:
8970:
8963:
8956:10.2307/2589133
8938:
8934:
8922:
8918:
8903:10.2307/2299611
8885:
8881:
8870:
8866:
8854:
8850:
8833:
8827:
8823:
8806:
8800:
8796:
8779:
8773:
8764:
8756:
8752:
8735:
8729:
8725:
8708:
8702:
8693:
8678:
8674:
8657:
8651:
8647:
8629:
8625:
8614:
8610:
8603:10.2307/2322549
8585:
8581:
8569:
8562:
8551:
8532:
8520:
8516:
8499:
8493:
8484:
8472:
8459:
8443:
8437:
8433:
8416:
8410:
8401:
8389:
8376:
8359:
8353:
8332:
8314:
8307:
8290:
8284:
8255:
8250:
8223:
8169:
8166:
8165:
8105:
8102:
8101:
8097:if and only if
8063:
8011:
7994:
7992:
7979:
7971:
7969:
7967:
7964:
7963:
7916:
7913:
7912:
7841:
7819:tangent lengths
7790:
7778:
7773:
7742:
7738:
7729:
7725:
7707:
7703:
7694:
7690:
7680:
7670:
7666:
7657:
7653:
7635:
7631:
7622:
7618:
7608:
7606:
7591:
7587:
7578:
7574:
7556:
7552:
7543:
7539:
7529:
7519:
7515:
7506:
7502:
7484:
7480:
7471:
7467:
7457:
7455:
7453:
7450:
7449:
7414:
7410:
7401:
7397:
7379:
7375:
7366:
7362:
7358:
7342:
7338:
7329:
7325:
7307:
7303:
7294:
7290:
7286:
7284:
7263:
7259:
7250:
7246:
7228:
7224:
7215:
7211:
7207:
7191:
7187:
7178:
7174:
7156:
7152:
7143:
7139:
7135:
7133:
7131:
7128:
7127:
7104:
7100:
7094:
7090:
7078:
7074:
7068:
7064:
7052:
7048:
7042:
7038:
7026:
7022:
7016:
7012:
7007:
7004:
7003:
6998:
6987:
6968:
6957:
6893:
6888:
6863:
6858:
6833:
6828:
6803:
6798:
6796:
6793:
6792:
6756:
6752:
6747:
6736:
6732:
6727:
6716:
6712:
6707:
6696:
6692:
6687:
6685:
6682:
6681:
6651:
6644:
6637:
6630:
6555:
6551:
6542:
6538:
6529:
6525:
6516:
6512:
6510:
6507:
6506:
6477:
6470:
6463:
6456:
6426:
6422:
6417:
6406:
6402:
6397:
6386:
6382:
6377:
6366:
6362:
6357:
6355:
6352:
6351:
6342:
6333:
6324:
6315:
6280:
6276:
6271:
6260:
6256:
6251:
6240:
6236:
6231:
6220:
6216:
6211:
6209:
6206:
6205:
6196:
6189:
6182:
6175:
6145:
6141:
6136:
6125:
6121:
6116:
6105:
6101:
6096:
6085:
6081:
6076:
6074:
6071:
6070:
6046:
6039:
6032:
6025:
5982:
5950:
5934:
5932:
5929:
5928:
5860:
5849:
5847:
5825:
5808:
5806:
5794:
5790:
5786:
5779:
5775:
5771:
5769:
5767:
5764:
5763:
5737:
5730:
5674:
5671:
5670:
5598:
5560:
5557:
5556:
5518:
5492:
5490:
5481:
5477:
5454:
5451:
5450:
5403:
5386:
5384:
5371:
5363:
5361:
5337:
5320:
5318:
5305:
5297:
5295:
5293:
5290:
5289:
5272:
5224:
5218:are collinear.
5216:
5209:
5202:
5195:
5164:
5157:
5150:
5143:
5102:"area centroid"
5090:
5086:
5079:
5075:
5064:
5057:
5046:
5039:
5032:
5025:
5018:
5011:
5004:
4997:
4957:are collinear.
4912:
4901:
4894:
4883:
4848:
4841:
4828:of the segment
4827:
4821:and the middle
4820:
4813:
4797:
4766:
4758:
4755:
4753:
4750:
4749:
4727:
4719:
4716:
4714:
4711:
4710:
4581:
4573:
4571:
4560:
4556:
4550:
4546:
4544:
4542:
4539:
4538:
4344:
4342:
4333:
4330:
4329:
4193:
4191:
4182:
4179:
4178:
4172:tangency chords
4168:tangent lengths
4148:
4146:Tangency chords
4120:
4119:
4071:
4070:
4057:
4046:
4044:
4042:
4033:
4030:
4029:
4005:
4004:
3956:
3955:
3942:
3931:
3929:
3927:
3918:
3915:
3914:
3880:tangent lengths
3860:
3776:
3696:
3693:
3685:
3677:
3674:
3673:
3660:tangency chords
3592:
3545:
3542:
3529:
3521:
3518:
3517:
3452:
3405:
3402:
3389:
3381:
3378:
3377:
3312:
3265:
3262:
3249:
3241:
3238:
3237:
3172:
3125:
3122:
3109:
3101:
3098:
3097:
3067:tangent lengths
3047:
3025:
3021:
3015:
3011:
3005:
3001:
2992:
2988:
2982:
2978:
2972:
2968:
2959:
2955:
2949:
2945:
2939:
2935:
2926:
2922:
2916:
2912:
2906:
2902:
2894:
2891:
2890:
2864:
2860:
2854:
2850:
2841:
2837:
2831:
2827:
2820:
2808:
2804:
2798:
2794:
2785:
2781:
2775:
2771:
2762:
2758:
2752:
2748:
2742:
2738:
2732:
2728:
2716:
2712:
2710:
2703:
2701:
2693:
2690:
2689:
2665:
2661:
2652:
2648:
2639:
2635:
2626:
2622:
2620:
2617:
2616:
2523:
2515:
2512:
2511:
2417:
2331:
2328:
2317:
2314:
2313:
2242:
2195:
2192:
2183:
2180:
2179:
2173:tangent lengths
2122:
2117:
2101:
2096:
2083:
2075:
2072:
2071:
2049:
2007:
1982:
1978:
1967:
1964:
1963:
1956:, there is the
1913:
1910:
1909:
1865:
1857:
1854:
1853:
1831:
1796:
1791:
1759:
1747:
1739:
1736:
1735:
1699:
1697:
1681:
1665:
1645:
1642:
1641:
1601:
1599:
1554:
1550:
1542:
1539:
1538:
1493:
1485:
1482:
1481:
1435:
1433:
1411:
1392:
1390:
1368:
1359:
1356:
1355:
1329:
1297:
1295:
1292:
1291:
1258:
1254:
1219:
1211:
1208:
1207:
1096:
1087:
1084:
1083:
1063:. If these are
1061:tangent lengths
1010:
1006:
979:
975:
969:
965:
963:
951:
942:
939:
938:
883:
880:
879:
866:
861:
841:tangency chords
810:tangent lengths
783:
753:
744:
735:
726:
699:
695:
689:
685:
676:
672:
666:
662:
660:
657:
656:
593:
591:
563:
561:
533:
531:
503:
501:
493:
490:
489:
474:to each other.
398:
395:
394:
389:
339:
336:
335:
319:) intersect at
315:(that is not a
242:
240:
214:
211:
210:
190:angle bisectors
186:
138:
17:
12:
11:
5:
9705:
9695:
9694:
9677:
9676:
9674:
9673:
9668:
9663:
9658:
9653:
9648:
9643:
9638:
9633:
9631:Pseudotriangle
9628:
9623:
9618:
9613:
9608:
9603:
9598:
9593:
9588:
9582:
9580:
9576:
9575:
9573:
9572:
9567:
9562:
9557:
9552:
9547:
9542:
9537:
9531:
9529:
9522:
9521:
9518:
9517:
9515:
9514:
9509:
9504:
9499:
9494:
9489:
9484:
9479:
9474:
9468:
9466:
9462:
9461:
9459:
9458:
9453:
9448:
9443:
9438:
9433:
9428:
9423:
9421:Dodecagon (12)
9418:
9412:
9410:
9406:
9405:
9403:
9402:
9397:
9392:
9387:
9382:
9377:
9372:
9367:
9362:
9357:
9351:
9349:
9342:
9336:
9335:
9333:
9332:
9327:
9322:
9317:
9312:
9307:
9302:
9297:
9292:
9287:
9282:
9277:
9272:
9267:
9262:
9257:
9252:
9247:
9242:
9236:
9234:
9232:Quadrilaterals
9228:
9227:
9225:
9224:
9219:
9214:
9209:
9204:
9199:
9194:
9188:
9186:
9180:
9179:
9167:
9166:
9159:
9152:
9144:
9138:
9137:
9116:
9115:External links
9113:
9110:
9109:
9082:
9055:
9033:
9006:
8979:
8961:
8950:(7): 657–658,
8932:
8916:
8897:(9): 462–465,
8879:
8864:
8848:
8821:
8794:
8762:
8750:
8723:
8691:
8672:
8645:
8623:
8608:
8579:
8560:
8530:
8514:
8482:
8457:
8431:
8399:
8374:
8330:
8305:
8252:
8251:
8249:
8246:
8245:
8244:
8239:
8234:
8229:
8222:
8219:
8218:
8217:
8206:
8203:
8200:
8197:
8194:
8191:
8188:
8185:
8182:
8179:
8176:
8173:
8151:
8150:
8139:
8136:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8062:
8059:
8041:
8040:
8026:
8023:
8020:
8017:
8014:
8009:
8006:
8003:
8000:
7997:
7991:
7985:
7982:
7977:
7974:
7961:
7950:
7947:
7944:
7941:
7938:
7935:
7932:
7929:
7926:
7923:
7920:
7910:
7840:
7837:
7836:
7835:
7832:
7829:
7822:
7815:
7812:
7805:
7789:
7786:
7777:
7774:
7772:
7769:
7768:
7767:
7756:
7750:
7745:
7741:
7737:
7732:
7728:
7724:
7721:
7718:
7715:
7710:
7706:
7702:
7697:
7693:
7689:
7686:
7683:
7678:
7673:
7669:
7665:
7660:
7656:
7652:
7649:
7646:
7643:
7638:
7634:
7630:
7625:
7621:
7617:
7614:
7611:
7605:
7599:
7594:
7590:
7586:
7581:
7577:
7573:
7570:
7567:
7564:
7559:
7555:
7551:
7546:
7542:
7538:
7535:
7532:
7527:
7522:
7518:
7514:
7509:
7505:
7501:
7498:
7495:
7492:
7487:
7483:
7479:
7474:
7470:
7466:
7463:
7460:
7443:
7442:
7428:
7425:
7422:
7417:
7413:
7409:
7404:
7400:
7396:
7393:
7390:
7387:
7382:
7378:
7374:
7369:
7365:
7361:
7356:
7353:
7350:
7345:
7341:
7337:
7332:
7328:
7324:
7321:
7318:
7315:
7310:
7306:
7302:
7297:
7293:
7289:
7283:
7277:
7274:
7271:
7266:
7262:
7258:
7253:
7249:
7245:
7242:
7239:
7236:
7231:
7227:
7223:
7218:
7214:
7210:
7205:
7202:
7199:
7194:
7190:
7186:
7181:
7177:
7173:
7170:
7167:
7164:
7159:
7155:
7151:
7146:
7142:
7138:
7121:
7120:
7107:
7103:
7097:
7093:
7089:
7086:
7081:
7077:
7071:
7067:
7063:
7060:
7055:
7051:
7045:
7041:
7037:
7034:
7029:
7025:
7019:
7015:
7011:
6996:
6985:
6977:into segments
6966:
6955:
6926:
6925:
6911:
6908:
6905:
6902:
6899:
6896:
6892:
6887:
6881:
6878:
6875:
6872:
6869:
6866:
6862:
6857:
6851:
6848:
6845:
6842:
6839:
6836:
6832:
6827:
6821:
6818:
6815:
6812:
6809:
6806:
6802:
6778:
6777:
6766:
6759:
6755:
6751:
6746:
6739:
6735:
6731:
6726:
6719:
6715:
6711:
6706:
6699:
6695:
6691:
6649:
6642:
6635:
6628:
6575:
6574:
6563:
6558:
6554:
6550:
6545:
6541:
6537:
6532:
6528:
6524:
6519:
6515:
6475:
6468:
6461:
6454:
6448:
6447:
6436:
6429:
6425:
6421:
6416:
6409:
6405:
6401:
6396:
6389:
6385:
6381:
6376:
6369:
6365:
6361:
6338:
6329:
6320:
6311:
6302:
6301:
6290:
6283:
6279:
6275:
6270:
6263:
6259:
6255:
6250:
6243:
6239:
6235:
6230:
6223:
6219:
6215:
6194:
6187:
6180:
6173:
6167:
6166:
6155:
6148:
6144:
6140:
6135:
6128:
6124:
6120:
6115:
6108:
6104:
6100:
6095:
6088:
6084:
6080:
6065:if and only if
6044:
6037:
6030:
6023:
5981:
5978:
5957:
5953:
5947:
5944:
5941:
5938:
5884:
5883:
5869:
5866:
5863:
5858:
5855:
5852:
5846:
5840:
5837:
5834:
5831:
5828:
5823:
5820:
5817:
5814:
5811:
5805:
5797:
5793:
5789:
5782:
5778:
5774:
5755:with incenter
5735:
5728:
5723:
5722:
5711:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5681:
5678:
5665:if and only if
5649:
5648:
5637:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5605:
5602:
5597:
5594:
5591:
5588:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5542:
5541:
5530:
5524:
5521:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5489:
5484:
5480:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5442:with incenter
5436:
5435:
5424:
5418:
5415:
5412:
5409:
5406:
5401:
5398:
5395:
5392:
5389:
5383:
5377:
5374:
5369:
5366:
5358:
5352:
5349:
5346:
5343:
5340:
5335:
5332:
5329:
5326:
5323:
5317:
5311:
5308:
5303:
5300:
5271:
5268:
5223:
5220:
5214:
5207:
5200:
5193:
5162:
5155:
5148:
5141:
5130:with incenter
5088:
5084:
5077:
5073:
5062:
5055:
5044:
5037:
5030:
5023:
5016:
5009:
5002:
4995:
4910:
4899:
4892:
4881:
4866:with incenter
4846:
4839:
4825:
4818:
4811:
4796:
4793:
4772:
4769:
4764:
4761:
4733:
4730:
4725:
4722:
4632:
4631:
4624:
4605:
4604:
4593:
4587:
4584:
4579:
4576:
4570:
4563:
4559:
4553:
4549:
4476:
4475:
4460:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4403:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4341:
4338:
4327:
4315:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4190:
4187:
4147:
4144:
4143:
4142:
4130:
4123:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4074:
4066:
4063:
4060:
4055:
4052:
4049:
4041:
4038:
4027:
4015:
4008:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3959:
3951:
3948:
3945:
3940:
3937:
3934:
3926:
3923:
3859:
3856:
3855:
3854:
3843:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3692:
3688:
3684:
3681:
3656:
3655:
3644:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3541:
3536:
3533:
3528:
3525:
3515:
3504:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3401:
3396:
3393:
3388:
3385:
3375:
3364:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3261:
3256:
3253:
3248:
3245:
3235:
3224:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3121:
3116:
3113:
3108:
3105:
3046:
3045:Angle formulas
3043:
3028:
3024:
3018:
3014:
3008:
3004:
3000:
2995:
2991:
2985:
2981:
2975:
2971:
2967:
2962:
2958:
2952:
2948:
2942:
2938:
2934:
2929:
2925:
2919:
2915:
2909:
2905:
2901:
2898:
2887:
2886:
2872:
2867:
2863:
2857:
2853:
2849:
2844:
2840:
2834:
2830:
2826:
2823:
2816:
2811:
2807:
2801:
2797:
2793:
2788:
2784:
2778:
2774:
2770:
2765:
2761:
2755:
2751:
2745:
2741:
2735:
2731:
2727:
2724:
2719:
2715:
2709:
2706:
2700:
2697:
2668:
2664:
2660:
2655:
2651:
2647:
2642:
2638:
2634:
2629:
2625:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2531:
2528:
2522:
2519:
2508:
2507:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2327:
2324:
2321:
2283:
2282:
2270:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2191:
2188:
2146:
2145:
2131:
2128:
2125:
2121:
2116:
2110:
2107:
2104:
2100:
2095:
2090:
2087:
2082:
2079:
2048:
2045:
2006:
2003:
1999:
1998:
1985:
1981:
1977:
1974:
1971:
1938:
1937:
1926:
1923:
1920:
1917:
1892:
1891:
1878:
1875:
1872:
1869:
1864:
1861:
1830:
1827:
1815:
1814:
1803:
1799:
1794:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1762:
1755:
1752:
1746:
1743:
1729:
1728:
1717:
1712:
1708:
1705:
1702:
1696:
1693:
1688:
1685:
1680:
1677:
1672:
1669:
1664:
1661:
1658:
1655:
1652:
1649:
1628:
1627:
1614:
1610:
1607:
1604:
1598:
1595:
1591:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1553:
1549:
1546:
1506:
1503:
1500:
1497:
1492:
1489:
1466:
1465:
1453:
1448:
1444:
1441:
1438:
1432:
1429:
1424:
1421:
1418:
1415:
1410:
1405:
1401:
1398:
1395:
1389:
1386:
1381:
1378:
1375:
1372:
1367:
1364:
1328:
1325:
1310:
1307:
1304:
1301:
1280:
1279:
1268:
1261:
1257:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1218:
1215:
1193:
1192:
1180:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1095:
1092:
1040:and the sides
1030:
1029:
1013:
1009:
1005:
1002:
999:
996:
993:
990:
987:
982:
978:
972:
968:
959:
956:
950:
947:
916:
915:
903:
900:
897:
894:
891:
888:
865:
862:
860:
857:
782:
779:
749:
740:
731:
722:
716:
715:
702:
698:
692:
688:
684:
679:
675:
669:
665:
626:
625:
614:
609:
605:
602:
599:
596:
590:
587:
584:
579:
575:
572:
569:
566:
560:
557:
554:
549:
545:
542:
539:
536:
530:
527:
524:
519:
515:
512:
509:
506:
500:
497:
449:
448:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
387:
386:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
329:if and only if
290:
289:
278:
275:
272:
267:
263:
260:
257:
254:
251:
248:
245:
239:
236:
233:
230:
227:
224:
221:
218:
185:
182:
137:
134:
124:. The section
15:
9:
6:
4:
3:
2:
9704:
9693:
9690:
9689:
9687:
9672:
9671:Weakly simple
9669:
9667:
9664:
9662:
9659:
9657:
9654:
9652:
9649:
9647:
9644:
9642:
9639:
9637:
9634:
9632:
9629:
9627:
9624:
9622:
9619:
9617:
9614:
9612:
9611:Infinite skew
9609:
9607:
9604:
9602:
9599:
9597:
9594:
9592:
9589:
9587:
9584:
9583:
9581:
9577:
9571:
9568:
9566:
9563:
9561:
9558:
9556:
9553:
9551:
9548:
9546:
9543:
9541:
9538:
9536:
9533:
9532:
9530:
9527:
9526:Star polygons
9523:
9513:
9512:Apeirogon (∞)
9510:
9508:
9505:
9503:
9500:
9498:
9495:
9493:
9490:
9488:
9485:
9483:
9480:
9478:
9475:
9473:
9470:
9469:
9467:
9463:
9457:
9456:Icosagon (20)
9454:
9452:
9449:
9447:
9444:
9442:
9439:
9437:
9434:
9432:
9429:
9427:
9424:
9422:
9419:
9417:
9414:
9413:
9411:
9407:
9401:
9398:
9396:
9393:
9391:
9388:
9386:
9383:
9381:
9378:
9376:
9373:
9371:
9368:
9366:
9363:
9361:
9358:
9356:
9353:
9352:
9350:
9346:
9343:
9337:
9331:
9328:
9326:
9323:
9321:
9318:
9316:
9313:
9311:
9308:
9306:
9303:
9301:
9298:
9296:
9293:
9291:
9290:Parallelogram
9288:
9286:
9285:Orthodiagonal
9283:
9281:
9278:
9276:
9273:
9271:
9268:
9266:
9265:Ex-tangential
9263:
9261:
9258:
9256:
9253:
9251:
9248:
9246:
9243:
9241:
9238:
9237:
9235:
9233:
9229:
9223:
9220:
9218:
9215:
9213:
9210:
9208:
9205:
9203:
9200:
9198:
9195:
9193:
9190:
9189:
9187:
9185:
9181:
9176:
9172:
9165:
9160:
9158:
9153:
9151:
9146:
9145:
9142:
9134:
9133:
9128:
9124:
9119:
9118:
9104:
9100:
9093:
9086:
9077:
9073:
9066:
9059:
9050:
9046:
9045:
9037:
9028:
9024:
9017:
9010:
9001:
8997:
8990:
8983:
8975:
8968:
8966:
8957:
8953:
8949:
8945:
8944:
8936:
8929:
8926:
8920:
8912:
8908:
8904:
8900:
8896:
8892:
8891:
8883:
8877:
8875:
8868:
8859:
8852:
8843:
8839:
8832:
8825:
8816:
8812:
8805:
8798:
8789:
8785:
8778:
8771:
8769:
8767:
8759:
8754:
8745:
8741:
8734:
8727:
8718:
8714:
8707:
8700:
8698:
8696:
8688:
8685:
8681:
8676:
8667:
8663:
8656:
8649:
8641:(4): 239, 242
8640:
8636:
8635:
8627:
8621:
8619:
8612:
8604:
8600:
8596:
8592:
8591:
8583:
8576:
8573:
8567:
8565:
8558:
8556:
8549:
8547:
8545:
8543:
8541:
8539:
8537:
8535:
8525:
8518:
8509:
8505:
8498:
8491:
8489:
8487:
8477:
8470:
8468:
8466:
8464:
8462:
8453:
8449:
8442:
8435:
8426:
8422:
8415:
8408:
8406:
8404:
8394:
8387:
8385:
8383:
8381:
8379:
8369:
8365:
8358:
8351:
8349:
8347:
8345:
8343:
8341:
8339:
8337:
8335:
8325:
8321:
8320:
8312:
8310:
8300:
8296:
8289:
8282:
8280:
8278:
8276:
8274:
8272:
8270:
8268:
8266:
8264:
8262:
8260:
8258:
8253:
8243:
8240:
8238:
8235:
8233:
8230:
8228:
8225:
8224:
8204:
8201:
8198:
8195:
8192:
8189:
8186:
8183:
8180:
8177:
8174:
8171:
8164:
8163:
8162:
8160:
8156:
8137:
8134:
8131:
8128:
8125:
8122:
8119:
8116:
8113:
8110:
8107:
8100:
8099:
8098:
8096:
8092:
8088:
8084:
8080:
8076:
8072:
8068:
8058:
8055:
8053:
8049:
8046:
8024:
8021:
8018:
8015:
8012:
8007:
8004:
8001:
7998:
7995:
7989:
7983:
7980:
7975:
7972:
7962:
7948:
7945:
7942:
7939:
7936:
7933:
7930:
7927:
7924:
7921:
7918:
7911:
7909:
7905:
7902:
7901:
7900:
7898:
7894:
7890:
7886:
7882:
7878:
7874:
7870:
7866:
7862:
7858:
7850:
7845:
7833:
7830:
7827:
7823:
7820:
7816:
7813:
7810:
7809:perpendicular
7806:
7803:
7799:
7798:
7797:
7795:
7785:
7783:
7754:
7743:
7739:
7735:
7730:
7726:
7722:
7719:
7708:
7704:
7700:
7695:
7691:
7687:
7684:
7671:
7667:
7663:
7658:
7654:
7650:
7647:
7636:
7632:
7628:
7623:
7619:
7615:
7612:
7603:
7592:
7588:
7584:
7579:
7575:
7571:
7568:
7557:
7553:
7549:
7544:
7540:
7536:
7533:
7520:
7516:
7512:
7507:
7503:
7499:
7496:
7485:
7481:
7477:
7472:
7468:
7464:
7461:
7448:
7447:
7446:
7423:
7420:
7415:
7411:
7407:
7402:
7398:
7388:
7385:
7380:
7376:
7372:
7367:
7363:
7351:
7348:
7343:
7339:
7335:
7330:
7326:
7316:
7313:
7308:
7304:
7300:
7295:
7291:
7281:
7272:
7269:
7264:
7260:
7256:
7251:
7247:
7237:
7234:
7229:
7225:
7221:
7216:
7212:
7200:
7197:
7192:
7188:
7184:
7179:
7175:
7165:
7162:
7157:
7153:
7149:
7144:
7140:
7126:
7125:
7124:
7105:
7101:
7095:
7091:
7087:
7084:
7079:
7075:
7069:
7065:
7061:
7058:
7053:
7049:
7043:
7039:
7035:
7032:
7027:
7023:
7017:
7013:
7009:
7002:
7001:
7000:
6995:
6991:
6984:
6980:
6976:
6972:
6965:
6961:
6954:
6950:
6946:
6942:
6937:
6935:
6931:
6906:
6903:
6900:
6890:
6885:
6876:
6873:
6870:
6860:
6855:
6846:
6843:
6840:
6830:
6825:
6816:
6813:
6810:
6800:
6791:
6790:
6789:
6787:
6783:
6764:
6757:
6753:
6749:
6744:
6737:
6733:
6729:
6724:
6717:
6713:
6709:
6704:
6697:
6693:
6689:
6680:
6679:
6678:
6676:
6672:
6668:
6664:
6660:
6656:
6652:
6645:
6638:
6631:
6624:
6620:
6616:
6612:
6608:
6604:
6600:
6596:
6591:
6589:
6585:
6581:
6561:
6556:
6552:
6548:
6543:
6539:
6535:
6530:
6526:
6522:
6517:
6513:
6505:
6504:
6503:
6501:
6497:
6493:
6489:
6485:
6482:of triangles
6481:
6480:circumcircles
6474:
6467:
6460:
6453:
6434:
6427:
6423:
6419:
6414:
6407:
6403:
6399:
6394:
6387:
6383:
6379:
6374:
6367:
6363:
6359:
6350:
6349:
6348:
6346:
6341:
6337:
6332:
6328:
6323:
6319:
6314:
6310:
6307:
6288:
6281:
6277:
6273:
6268:
6261:
6257:
6253:
6248:
6241:
6237:
6233:
6228:
6221:
6217:
6213:
6204:
6203:
6202:
6200:
6193:
6186:
6179:
6172:
6153:
6146:
6142:
6138:
6133:
6126:
6122:
6118:
6113:
6106:
6102:
6098:
6093:
6086:
6082:
6078:
6069:
6068:
6067:
6066:
6062:
6058:
6054:
6050:
6043:
6036:
6029:
6022:
6017:
6015:
6011:
6007:
6003:
5999:
5995:
5986:
5977:
5975:
5971:
5955:
5951:
5945:
5942:
5939:
5936:
5925:
5920:
5917:
5913:
5909:
5905:
5901:
5897:
5893:
5889:
5867:
5864:
5861:
5856:
5853:
5850:
5844:
5838:
5835:
5832:
5829:
5826:
5821:
5818:
5815:
5812:
5809:
5803:
5795:
5791:
5787:
5780:
5776:
5772:
5762:
5761:
5760:
5758:
5754:
5750:
5746:
5742:
5738:
5731:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5676:
5669:
5668:
5667:
5666:
5662:
5658:
5654:
5651:The incenter
5635:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5603:
5600:
5595:
5592:
5589:
5586:
5583:
5580:
5577:
5574:
5571:
5568:
5565:
5562:
5555:
5554:
5553:
5551:
5547:
5528:
5522:
5519:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5493:
5487:
5482:
5478:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5449:
5448:
5447:
5445:
5441:
5422:
5416:
5413:
5410:
5407:
5404:
5399:
5396:
5393:
5390:
5387:
5381:
5375:
5372:
5367:
5364:
5356:
5350:
5347:
5344:
5341:
5338:
5333:
5330:
5327:
5324:
5321:
5315:
5309:
5306:
5301:
5298:
5288:
5287:
5286:
5284:
5279:
5277:
5267:
5265:
5261:
5257:
5253:
5249:
5245:
5240:
5237:
5236:conic section
5233:
5229:
5219:
5217:
5210:
5203:
5196:
5189:
5185:
5181:
5177:
5173:
5170:of triangles
5169:
5165:
5158:
5151:
5144:
5137:
5133:
5129:
5124:
5122:
5118:
5114:
5110:
5106:
5103:
5099:
5095:
5091:
5080:
5069:
5065:
5058:
5051:
5047:
5040:
5033:
5026:
5019:
5012:
5005:
4998:
4991:
4987:
4983:
4979:
4971:
4967:
4962:
4958:
4956:
4952:
4948:
4944:
4940:
4936:
4932:
4928:
4923:
4921:
4917:
4913:
4906:
4902:
4895:
4888:
4884:
4877:
4873:
4869:
4865:
4861:
4857:
4853:
4849:
4842:
4831:
4824:
4817:
4810:
4806:
4801:
4792:
4790:
4770:
4767:
4762:
4759:
4731:
4728:
4723:
4720:
4708:
4704:
4700:
4696:
4692:
4688:
4684:
4680:
4675:
4673:
4669:
4665:
4661:
4657:
4653:
4649:
4645:
4641:
4637:
4629:
4625:
4622:
4618:
4614:
4613:perpendicular
4610:
4609:
4608:
4591:
4585:
4582:
4577:
4574:
4568:
4561:
4557:
4551:
4547:
4537:
4536:
4535:
4533:
4529:
4525:
4521:
4517:
4513:
4509:
4505:
4501:
4497:
4493:
4489:
4485:
4481:
4455:
4452:
4449:
4440:
4437:
4434:
4425:
4422:
4419:
4410:
4407:
4404:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4345:
4339:
4336:
4328:
4313:
4304:
4301:
4298:
4289:
4286:
4283:
4274:
4271:
4268:
4259:
4256:
4253:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4194:
4188:
4185:
4177:
4176:
4175:
4173:
4169:
4165:
4161:
4157:
4153:
4128:
4116:
4113:
4110:
4107:
4101:
4098:
4095:
4086:
4083:
4080:
4064:
4061:
4058:
4053:
4050:
4047:
4039:
4036:
4028:
4013:
4001:
3998:
3995:
3992:
3986:
3983:
3980:
3971:
3968:
3965:
3949:
3946:
3943:
3938:
3935:
3932:
3924:
3921:
3913:
3912:
3911:
3909:
3905:
3901:
3897:
3893:
3889:
3885:
3881:
3877:
3873:
3869:
3865:
3841:
3831:
3828:
3825:
3816:
3813:
3810:
3801:
3798:
3795:
3786:
3783:
3780:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3690:
3686:
3682:
3679:
3672:
3671:
3670:
3668:
3664:
3661:
3642:
3632:
3629:
3626:
3617:
3614:
3611:
3602:
3599:
3596:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3539:
3534:
3531:
3526:
3523:
3516:
3502:
3492:
3489:
3486:
3477:
3474:
3471:
3462:
3459:
3456:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3399:
3394:
3391:
3386:
3383:
3376:
3362:
3352:
3349:
3346:
3337:
3334:
3331:
3322:
3319:
3316:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3259:
3254:
3251:
3246:
3243:
3236:
3222:
3212:
3209:
3206:
3197:
3194:
3191:
3182:
3179:
3176:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3119:
3114:
3111:
3106:
3103:
3096:
3095:
3094:
3092:
3088:
3084:
3080:
3076:
3072:
3068:
3064:
3060:
3056:
3052:
3042:
3026:
3022:
3016:
3012:
3006:
3002:
2998:
2993:
2989:
2983:
2979:
2973:
2969:
2965:
2960:
2956:
2950:
2946:
2940:
2936:
2932:
2927:
2923:
2917:
2913:
2907:
2903:
2899:
2896:
2865:
2861:
2855:
2851:
2847:
2842:
2838:
2832:
2828:
2821:
2809:
2805:
2799:
2795:
2791:
2786:
2782:
2776:
2772:
2763:
2759:
2753:
2749:
2743:
2739:
2733:
2729:
2725:
2722:
2717:
2713:
2707:
2704:
2698:
2695:
2688:
2687:
2686:
2684:
2666:
2662:
2658:
2653:
2649:
2645:
2640:
2636:
2632:
2627:
2623:
2614:
2610:
2606:
2602:
2597:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2529:
2526:
2520:
2517:
2487:
2484:
2481:
2478:
2475:
2466:
2463:
2460:
2457:
2454:
2445:
2442:
2439:
2436:
2433:
2427:
2424:
2421:
2418:
2410:
2407:
2404:
2401:
2398:
2389:
2386:
2383:
2380:
2377:
2368:
2365:
2362:
2359:
2356:
2347:
2344:
2341:
2338:
2335:
2325:
2322:
2319:
2312:
2311:
2310:
2308:
2304:
2300:
2296:
2292:
2288:
2268:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2189:
2186:
2178:
2177:
2176:
2174:
2169:
2167:
2164:(and hence a
2163:
2159:
2155:
2151:
2129:
2126:
2123:
2119:
2114:
2108:
2105:
2102:
2098:
2093:
2088:
2085:
2080:
2077:
2070:
2069:
2068:
2066:
2062:
2058:
2054:
2044:
2042:
2038:
2034:
2030:
2025:
2023:
2016:
2011:
2002:
1983:
1979:
1975:
1972:
1969:
1962:
1961:
1960:
1959:
1955:
1951:
1947:
1943:
1924:
1921:
1918:
1915:
1908:
1907:
1906:
1904:
1899:
1897:
1876:
1873:
1870:
1867:
1862:
1859:
1852:
1851:
1850:
1848:
1844:
1840:
1836:
1826:
1824:
1820:
1801:
1792:
1788:
1785:
1779:
1776:
1773:
1770:
1767:
1753:
1750:
1744:
1741:
1734:
1733:
1732:
1715:
1710:
1706:
1703:
1700:
1694:
1691:
1686:
1683:
1678:
1675:
1670:
1667:
1662:
1659:
1656:
1653:
1650:
1647:
1640:
1639:
1638:
1635:
1633:
1612:
1608:
1605:
1602:
1596:
1593:
1589:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1551:
1547:
1544:
1537:
1536:
1535:
1533:
1528:
1526:
1522:
1504:
1501:
1498:
1495:
1490:
1487:
1479:
1475:
1471:
1451:
1446:
1442:
1439:
1436:
1430:
1427:
1422:
1419:
1416:
1413:
1408:
1403:
1399:
1396:
1393:
1387:
1384:
1379:
1376:
1373:
1370:
1365:
1362:
1354:
1353:
1352:
1350:
1346:
1342:
1338:
1334:
1333:trigonometric
1324:
1308:
1305:
1302:
1299:
1289:
1285:
1266:
1259:
1251:
1248:
1245:
1242:
1239:
1233:
1230:
1227:
1224:
1221:
1216:
1213:
1206:
1205:
1204:
1202:
1198:
1178:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1093:
1090:
1082:
1081:
1080:
1078:
1074:
1070:
1066:
1062:
1057:
1055:
1051:
1047:
1043:
1039:
1035:
1011:
1003:
1000:
997:
994:
991:
985:
980:
976:
970:
966:
957:
954:
948:
945:
937:
936:
935:
933:
929:
925:
924:semiperimeter
921:
901:
898:
895:
892:
889:
886:
878:
877:
876:
874:
871:
856:
854:
850:
846:
842:
837:
835:
831:
827:
823:
819:
815:
811:
806:
804:
800:
792:
787:
778:
776:
771:
769:
765:
761:
757:
752:
748:
743:
739:
734:
730:
725:
721:
700:
696:
690:
686:
682:
677:
673:
667:
663:
655:
654:
653:
651:
647:
643:
639:
630:
612:
607:
603:
600:
597:
588:
585:
582:
577:
573:
570:
567:
558:
555:
552:
547:
543:
540:
537:
528:
525:
522:
517:
513:
510:
507:
498:
495:
488:
487:
486:
484:
480:
475:
473:
469:
465:
461:
453:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
393:
392:
391:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
334:
333:
332:
330:
326:
322:
318:
314:
309:
307:
303:
299:
295:
276:
273:
270:
265:
261:
258:
255:
252:
249:
246:
243:
237:
234:
231:
228:
225:
222:
219:
216:
209:
208:
207:
205:
202:
201:semiperimeter
198:
197:Pitot theorem
193:
191:
181:
179:
175:
171:
167:
163:
159:
155:
154:orthodiagonal
151:
147:
143:
136:Special cases
133:
131:
127:
123:
119:
114:
112:
108:
104:
100:
96:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
48:quadrilateral
46:
42:
38:
34:
30:
21:
9465:>20 sides
9400:Decagon (10)
9385:Heptagon (7)
9375:Pentagon (5)
9365:Triangle (3)
9319:
9260:Equidiagonal
9130:
9102:
9098:
9085:
9075:
9071:
9058:
9048:
9042:
9036:
9026:
9022:
9009:
8999:
8995:
8982:
8973:
8947:
8941:
8935:
8924:
8919:
8894:
8888:
8882:
8873:
8867:
8857:
8851:
8841:
8837:
8824:
8814:
8810:
8797:
8787:
8783:
8753:
8743:
8739:
8726:
8716:
8712:
8684:Cut-the-knot
8683:
8675:
8665:
8661:
8648:
8638:
8632:
8626:
8617:
8611:
8597:(1): 54–56,
8594:
8588:
8582:
8571:
8554:
8524:Trigonometry
8523:
8517:
8507:
8503:
8475:
8451:
8447:
8434:
8424:
8420:
8392:
8367:
8363:
8323:
8317:
8298:
8294:
8158:
8154:
8152:
8094:
8090:
8082:
8078:
8074:
8070:
8066:
8064:
8056:
8047:
8044:
8042:
7907:
7903:
7888:
7884:
7880:
7876:
7872:
7868:
7864:
7860:
7856:
7854:
7848:
7791:
7779:
7444:
7122:
6993:
6989:
6982:
6978:
6974:
6970:
6963:
6959:
6952:
6948:
6944:
6940:
6938:
6933:
6929:
6927:
6785:
6781:
6779:
6674:
6670:
6666:
6662:
6658:
6654:
6647:
6640:
6633:
6626:
6622:
6618:
6614:
6610:
6606:
6602:
6598:
6594:
6592:
6576:
6499:
6495:
6491:
6487:
6483:
6472:
6465:
6458:
6451:
6449:
6339:
6335:
6330:
6326:
6321:
6317:
6312:
6308:
6303:
6191:
6184:
6177:
6170:
6168:
6060:
6056:
6052:
6048:
6041:
6034:
6027:
6020:
6018:
6013:
6009:
6005:
6001:
5997:
5993:
5991:
5973:
5969:
5921:
5915:
5911:
5907:
5903:
5899:
5895:
5891:
5887:
5885:
5756:
5752:
5748:
5744:
5733:
5726:
5724:
5656:
5652:
5650:
5549:
5545:
5543:
5443:
5439:
5437:
5282:
5280:
5273:
5263:
5259:
5255:
5251:
5247:
5243:
5241:
5225:
5212:
5205:
5198:
5191:
5187:
5183:
5179:
5175:
5171:
5168:orthocenters
5160:
5153:
5146:
5139:
5135:
5131:
5127:
5125:
5120:
5116:
5112:
5108:
5104:
5097:
5082:
5071:
5067:
5060:
5053:
5042:
5035:
5028:
5021:
5014:
5007:
5000:
4993:
4989:
4985:
4981:
4977:
4975:
4969:
4965:
4954:
4950:
4946:
4942:
4938:
4934:
4930:
4926:
4924:
4908:
4904:
4897:
4890:
4886:
4879:
4875:
4871:
4867:
4863:
4859:
4855:
4844:
4837:
4835:
4829:
4822:
4815:
4808:
4804:
4788:
4706:
4702:
4698:
4694:
4690:
4686:
4682:
4678:
4676:
4671:
4667:
4663:
4659:
4651:
4647:
4643:
4639:
4635:
4633:
4617:circumcircle
4606:
4531:
4527:
4523:
4519:
4515:
4511:
4507:
4503:
4499:
4495:
4491:
4487:
4483:
4479:
4477:
4163:
4159:
4155:
4151:
4149:
3907:
3903:
3899:
3895:
3891:
3887:
3883:
3875:
3871:
3867:
3863:
3861:
3669:is given by
3666:
3662:
3657:
3086:
3082:
3078:
3074:
3070:
3062:
3058:
3054:
3050:
3048:
2888:
2685:is given by
2682:
2612:
2608:
2604:
2600:
2598:
2509:
2306:
2302:
2298:
2294:
2290:
2286:
2284:
2170:
2153:
2149:
2147:
2067:is given by
2064:
2060:
2056:
2052:
2050:
2026:
2019:
2014:
2000:
1953:
1949:
1941:
1939:
1902:
1900:
1893:
1846:
1842:
1838:
1834:
1832:
1829:Inequalities
1822:
1818:
1816:
1730:
1636:
1631:
1629:
1531:
1529:
1476:and hence a
1467:
1348:
1344:
1340:
1336:
1330:
1287:
1283:
1281:
1200:
1196:
1194:
1076:
1072:
1068:
1064:
1058:
1053:
1049:
1045:
1041:
1037:
1033:
1031:
927:
919:
917:
872:
867:
848:
844:
840:
838:
825:
821:
817:
813:
809:
807:
802:
798:
796:
790:
772:
767:
763:
759:
755:
750:
746:
741:
737:
732:
728:
723:
719:
717:
649:
645:
641:
637:
635:
482:
478:
476:
467:
463:
459:
457:
388:
324:
320:
312:
310:
305:
301:
297:
293:
291:
203:
194:
187:
162:circumcircle
139:
115:
106:
102:
98:
94:
90:
88:
79:
75:
71:
67:
63:
54:to a single
40:
36:
32:
26:
9661:Star-shaped
9636:Rectilinear
9606:Equilateral
9601:Equiangular
9565:Hendecagram
9409:11–20 sides
9390:Octagon (8)
9380:Hexagon (6)
9355:Monogon (1)
9197:Equilateral
8976:(6): 27–28.
8570:Yiu, Paul,
7895:(and hence
6197:denote the
5276:Newton line
5068:Nagel point
4920:Newton line
3089:, then the
2615:have radii
2031:with equal
2022:right kites
9666:Tangential
9570:Dodecagram
9348:1–10 sides
9339:By number
9320:Tangential
9300:Right kite
8248:References
8085:is also a
5446:satisfies
5228:concurrent
5121:Nagel line
2037:perimeters
2035:and equal
1958:inequality
1849:satisfies
1201:e, f, g, h
1197:a, b, c, d
808:The eight
331:either of
158:right kite
9646:Reinhardt
9555:Enneagram
9545:Heptagram
9535:Pentagram
9502:65537-gon
9360:Digon (2)
9330:Trapezoid
9295:Rectangle
9245:Bicentric
9207:Isosceles
9184:Triangles
9132:MathWorld
9105:: 381–385
9078:: 389–396
9029:: 211–212
8844:: 245–254
8817:: 289–295
8790:: 165–173
8746:: 155–164
8719:: 165–174
8510:: 103–106
8427:: 113–118
8370:: 119–130
8196:⋅
8178:⋅
8132:⋅
8114:⋅
8087:trapezoid
7943:⋅
7925:⋅
7897:bicentric
7826:bimedians
7802:diagonals
7723:−
7688:−
7664:−
7629:−
7572:−
7537:−
7513:−
7478:−
7349:−
7314:−
7198:−
7163:−
6895:△
6865:△
6835:△
6805:△
6584:excircles
6345:excircles
6199:altitudes
5833:⋅
5816:⋅
5741:midpoints
5701:⋅
5683:⋅
5625:⋅
5616:⋅
5607:⋅
5587:⋅
5569:⋅
5509:⋅
5500:⋅
5463:⋅
5411:⋅
5394:⋅
5345:⋅
5328:⋅
5094:perimeter
4916:collinear
4852:midpoints
4621:bicentric
3858:Diagonals
3687:φ
3683:
3527:
3387:
3247:
3107:
2723:−
2518:σ
2402:−
2399:σ
2381:−
2378:σ
2360:−
2357:σ
2339:−
2336:σ
1973:≥
1919:≥
1863:≤
1793:θ
1789:
1774:−
1695:
1679:
1663:
1597:
1580:⋅
1562:⋅
1431:
1388:
1246:−
1234:−
998:−
986:−
896:⋅
853:diagonals
834:congruent
595:∠
589:
583:⋅
565:∠
559:
535:∠
529:
523:⋅
505:∠
499:
426:−
408:−
317:trapezoid
174:trapezoid
122:rectangle
118:triangles
9686:Category
9621:Isotoxal
9616:Isogonal
9560:Decagram
9550:Octagram
9540:Hexagram
9341:of sides
9270:Harmonic
9171:Polygons
8616:Post at
8221:See also
7891:is also
6947:into as
6928:where ∆(
5759:, then
5739:are the
5270:Incenter
5048:are the
4850:are the
4656:bimedian
4166:are the
3878:are the
3065:are the
2047:Inradius
2041:incenter
2029:polygons
1525:calculus
932:inradius
839:The two
773:Several
68:inradius
64:incenter
60:incircle
9641:Regular
9586:Concave
9579:Classes
9487:257-gon
9310:Rhombus
9250:Crossed
9002:: 13–25
8911:2299611
8668:: 27–34
8454:: 63–77
8301:: 65–82
7782:rhombus
7776:Rhombus
6306:exradii
5970:a,b,c,d
5552:, then
5254:, then
5166:be the
4619:(it is
2309:, then
2158:maximum
1480:. Then
1470:maximum
930:is the
922:is the
472:tangent
150:squares
52:tangent
9651:Simple
9596:Cyclic
9591:Convex
9315:Square
9255:Cyclic
9217:Obtuse
9212:Kepler
8909:
8876:, 2011
8620:, 2012
8557:, 2008
8050:is an
7893:cyclic
6665:, and
6646:, and
6613:, and
6494:, and
6471:, and
6334:, and
6190:, and
6059:, and
6040:, and
5968:where
5886:where
5262:where
5138:, let
5100:, the
4907:, and
3908:q = BD
3904:p = AC
3091:angles
2889:where
2510:where
2307:y = DI
2303:x = CI
2299:v = BI
2295:u = AI
2162:cyclic
2148:where
1946:square
1940:where
1817:where
1630:where
1474:cyclic
1282:Since
918:where
830:vertex
718:where
166:cyclic
146:rhombi
105:, and
78:, and
56:circle
45:convex
9626:Magic
9222:Right
9202:Ideal
9192:Acute
9095:(PDF)
9068:(PDF)
9019:(PDF)
8992:(PDF)
8974:Kvant
8907:JSTOR
8834:(PDF)
8807:(PDF)
8780:(PDF)
8736:(PDF)
8709:(PDF)
8658:(PDF)
8500:(PDF)
8444:(PDF)
8417:(PDF)
8360:(PDF)
8291:(PDF)
5922:If a
4878:with
3882:from
2293:. If
2033:areas
142:kites
43:is a
39:) or
9656:Skew
9280:Kite
9175:List
8157:and
8153:and
8093:and
8083:ABCD
8077:and
8069:and
8048:WXYZ
7889:ABCD
7849:ABCD
7824:The
7794:kite
7788:Kite
6988:and
6958:and
6782:ABCD
6673:and
6621:and
6595:ABCD
6500:ABCD
6019:Let
6010:ABCD
5914:and
5898:and
5753:ABCD
5747:and
5732:and
5657:ABCD
5550:ABCD
5440:ABCD
5246:and
5128:ABCD
5081:and
4953:and
4937:and
4929:and
4914:are
4874:and
4864:ABCD
4858:and
4843:and
4814:and
4689:and
4679:ABCD
4670:and
4662:and
4650:and
4644:ABCD
4638:and
4628:kite
4611:are
4522:and
4494:and
4174:are
4162:and
3910:are
3906:and
3900:ABCD
3894:and
3874:and
3665:and
3087:ABCD
3081:and
3061:and
2683:ABCD
2305:and
2291:ABCD
1532:ABCD
926:and
870:area
868:The
859:Area
847:and
483:ABCD
470:are
466:and
460:ABCD
323:and
313:ABCD
156:. A
116:All
31:, a
8952:doi
8948:107
8899:doi
8599:doi
8073:at
7871:at
7445:or
7123:or
6934:APB
6930:APB
6667:DPA
6663:CPD
6659:BPC
6655:APB
6615:DPA
6611:CPD
6607:BPC
6603:APB
6496:DPA
6492:CPD
6488:BPC
6484:APB
6450:If
6061:DPA
6057:CPD
6053:BPC
6049:APB
6006:DPA
6002:CPD
5998:BPC
5994:APB
5725:If
5544:If
5184:DIA
5180:CID
5176:BIC
5172:AIB
5115:= 2
4992:at
4836:If
4705:at
4693:on
4685:on
4150:If
3862:If
3680:sin
3524:sin
3384:sin
3244:sin
3104:sin
3049:If
2613:DAB
2609:CDA
2605:BCD
2601:ABC
2168:).
1786:tan
1692:sin
1676:csc
1660:sin
1594:sin
1428:sin
1385:sin
1203:as
586:tan
556:tan
526:tan
496:tan
468:ADC
464:ABC
390:or
27:In
9688::
9129:,
9125:,
9103:14
9101:,
9097:,
9076:14
9074:,
9070:,
9049:95
9047:,
9027:11
9025:,
9021:,
9000:12
8998:,
8994:,
8964:^
8946:,
8927:,
8905:,
8895:33
8893:,
8842:17
8840:,
8836:,
8813:,
8809:,
8788:10
8786:,
8782:,
8765:^
8744:11
8742:,
8738:,
8717:11
8715:,
8711:,
8694:^
8686:,
8666:10
8664:,
8660:,
8639:57
8637:,
8595:93
8593:,
8574:,
8563:^
8533:^
8506:,
8502:,
8485:^
8460:^
8452:12
8450:,
8446:,
8423:,
8419:,
8402:^
8377:^
8368:10
8366:,
8362:,
8333:^
8324:94
8322:,
8308:^
8299:11
8297:,
8293:,
8256:^
8159:BC
8155:AD
8095:CD
8091:AB
8071:CD
8067:AB
8054:.
7908:XZ
7904:WY
7883:,
7879:,
7875:,
7869:DA
7867:,
7865:CD
7863:,
7861:BC
7859:,
7857:AB
6992:=
6990:PD
6981:=
6979:BP
6975:BD
6962:=
6960:PC
6951:=
6949:AP
6945:AC
6936:.
6661:,
6657:,
6639:,
6632:,
6609:,
6605:,
6590:.
6490:,
6486:,
6464:,
6457:,
6325:,
6316:,
6183:,
6176:,
6055:,
6051:,
6033:,
6026:,
6004:,
6000:,
5996:,
5910:,
5906:,
5894:,
5890:,
5749:BD
5745:AC
5260:IP
5256:JK
5211:,
5204:,
5197:,
5190:,
5182:,
5178:,
5174:,
5159:,
5152:,
5145:,
5117:GI
5113:NG
5061:BN
5059:=
5054:AT
5041:,
5034:,
5027:,
5013:,
5006:,
4999:,
4990:DA
4988:,
4986:CD
4984:,
4982:BC
4980:,
4978:AB
4949:,
4945:,
4903:,
4896:,
4887:JK
4860:BD
4856:AC
4830:JK
4791:.
4789:BD
4703:BD
4699:WY
4695:CD
4687:AB
4674:.
4672:DA
4668:BC
4664:CD
4660:AB
4652:DA
4648:BC
4640:CD
4636:AB
4623:).
4530:+
4526:=
4518:+
4514:=
4502:+
4498:=
4490:+
4486:=
4158:,
4154:,
3890:,
3886:,
3870:,
3866:,
3077:,
3073:,
3057:,
3053:,
3041:.
2611:,
2607:,
2603:,
2596:.
2301:,
2297:,
2063:,
2059:,
2055:,
2043:.
2024:.
1954:rs
1952:=
1898:.
1845:,
1841:,
1837:,
1527:.
1347:,
1343:,
1339:,
1331:A
1288:fh
1286:=
1284:eg
1075:,
1071:,
1067:,
1052:,
1048:,
1044:,
1036:,
824:,
820:,
816:,
766:,
762:,
758:,
745:,
736:,
727:,
648:,
644:,
640:,
479:BD
304:+
300:=
296:+
180:.
101:,
97:,
93:,
86:.
74:,
9177:)
9173:(
9163:e
9156:t
9149:v
9107:.
9080:.
9053:.
9031:.
9004:.
8959:.
8954::
8930:.
8914:.
8901::
8862:.
8846:.
8819:.
8815:6
8792:.
8748:.
8721:.
8689:.
8670:.
8643:.
8606:.
8601::
8528:.
8512:.
8508:8
8480:.
8429:.
8425:9
8397:.
8372:.
8328:.
8303:.
8205:.
8202:Y
8199:D
8193:Y
8190:C
8187:=
8184:W
8181:B
8175:W
8172:A
8138:Y
8135:C
8129:W
8126:B
8123:=
8120:Y
8117:D
8111:W
8108:A
8079:Y
8075:W
8025:Z
8022:D
8019:+
8016:X
8013:B
8008:Y
8005:C
8002:+
7999:W
7996:A
7990:=
7984:D
7981:B
7976:C
7973:A
7949:Y
7946:D
7940:W
7937:B
7934:=
7931:Y
7928:C
7922:W
7919:A
7885:Z
7881:Y
7877:X
7873:W
7811:.
7804:.
7755:.
7749:)
7744:2
7740:q
7736:+
7731:1
7727:p
7720:d
7717:(
7714:)
7709:1
7705:q
7701:+
7696:2
7692:p
7685:b
7682:(
7677:)
7672:2
7668:q
7659:1
7655:p
7651:+
7648:d
7645:(
7642:)
7637:1
7633:q
7624:2
7620:p
7616:+
7613:b
7610:(
7604:=
7598:)
7593:2
7589:q
7585:+
7580:2
7576:p
7569:c
7566:(
7563:)
7558:1
7554:q
7550:+
7545:1
7541:p
7534:a
7531:(
7526:)
7521:2
7517:q
7508:2
7504:p
7500:+
7497:c
7494:(
7491:)
7486:1
7482:q
7473:1
7469:p
7465:+
7462:a
7459:(
7427:)
7424:d
7421:+
7416:2
7412:q
7408:+
7403:1
7399:p
7395:(
7392:)
7389:b
7386:+
7381:1
7377:q
7373:+
7368:2
7364:p
7360:(
7355:)
7352:d
7344:2
7340:q
7336:+
7331:1
7327:p
7323:(
7320:)
7317:b
7309:1
7305:q
7301:+
7296:2
7292:p
7288:(
7282:=
7276:)
7273:c
7270:+
7265:2
7261:q
7257:+
7252:2
7248:p
7244:(
7241:)
7238:a
7235:+
7230:1
7226:q
7222:+
7217:1
7213:p
7209:(
7204:)
7201:c
7193:2
7189:q
7185:+
7180:2
7176:p
7172:(
7169:)
7166:a
7158:1
7154:q
7150:+
7145:1
7141:p
7137:(
7106:1
7102:q
7096:2
7092:p
7088:d
7085:+
7080:2
7076:q
7070:1
7066:p
7062:b
7059:=
7054:1
7050:q
7044:1
7040:p
7036:c
7033:+
7028:2
7024:q
7018:2
7014:p
7010:a
6997:2
6994:q
6986:1
6983:q
6971:P
6967:2
6964:p
6956:1
6953:p
6941:P
6910:)
6907:A
6904:P
6901:D
6898:(
6891:d
6886:+
6880:)
6877:C
6874:P
6871:B
6868:(
6861:b
6856:=
6850:)
6847:D
6844:P
6841:C
6838:(
6831:c
6826:+
6820:)
6817:B
6814:P
6811:A
6808:(
6801:a
6786:P
6765:.
6758:d
6754:R
6750:1
6745:+
6738:b
6734:R
6730:1
6725:=
6718:c
6714:R
6710:1
6705:+
6698:a
6694:R
6690:1
6675:D
6671:B
6650:d
6648:R
6643:c
6641:R
6636:b
6634:R
6629:a
6627:R
6623:D
6619:B
6599:P
6562:.
6557:4
6553:R
6549:+
6544:2
6540:R
6536:=
6531:3
6527:R
6523:+
6518:1
6514:R
6476:4
6473:R
6469:3
6466:R
6462:2
6459:R
6455:1
6452:R
6435:.
6428:d
6424:r
6420:1
6415:+
6408:b
6404:r
6400:1
6395:=
6388:c
6384:r
6380:1
6375:+
6368:a
6364:r
6360:1
6340:d
6336:r
6331:c
6327:r
6322:b
6318:r
6313:a
6309:r
6289:.
6282:4
6278:h
6274:1
6269:+
6262:2
6258:h
6254:1
6249:=
6242:3
6238:h
6234:1
6229:+
6222:1
6218:h
6214:1
6195:4
6192:h
6188:3
6185:h
6181:2
6178:h
6174:1
6171:h
6154:.
6147:4
6143:r
6139:1
6134:+
6127:2
6123:r
6119:1
6114:=
6107:3
6103:r
6099:1
6094:+
6087:1
6083:r
6079:1
6045:4
6042:r
6038:3
6035:r
6031:2
6028:r
6024:1
6021:r
6014:P
5974:s
5956:s
5952:/
5946:d
5943:c
5940:b
5937:a
5916:D
5912:C
5908:B
5904:A
5900:h
5896:g
5892:f
5888:e
5868:h
5865:+
5862:f
5857:g
5854:+
5851:e
5845:=
5839:D
5836:I
5830:B
5827:I
5822:C
5819:I
5813:A
5810:I
5804:=
5796:q
5792:M
5788:I
5781:p
5777:M
5773:I
5757:I
5736:q
5734:M
5729:p
5727:M
5710:.
5707:D
5704:I
5698:B
5695:I
5692:=
5689:C
5686:I
5680:A
5677:I
5653:I
5636:.
5631:A
5628:D
5622:D
5619:C
5613:C
5610:B
5604:B
5601:A
5596:=
5593:D
5590:I
5584:B
5581:I
5578:+
5575:C
5572:I
5566:A
5563:I
5546:I
5529:.
5523:D
5520:I
5515:C
5512:I
5506:B
5503:I
5497:A
5494:I
5488:+
5483:2
5479:B
5475:I
5472:=
5469:C
5466:B
5460:B
5457:A
5444:I
5423:.
5417:A
5414:I
5408:D
5405:I
5400:C
5397:I
5391:B
5388:I
5382:=
5376:A
5373:D
5368:C
5365:B
5357:,
5351:D
5348:I
5342:C
5339:I
5334:B
5331:I
5325:A
5322:I
5316:=
5310:D
5307:C
5302:B
5299:A
5283:I
5264:I
5252:P
5248:K
5244:J
5215:W
5213:H
5208:Z
5206:H
5201:Y
5199:H
5194:X
5192:H
5188:P
5163:W
5161:H
5156:Z
5154:H
5149:Y
5147:H
5142:X
5140:H
5136:P
5132:I
5109:I
5105:G
5098:N
5089:4
5087:N
5085:2
5083:N
5078:3
5076:N
5074:1
5072:N
5063:1
5056:1
5045:4
5043:N
5038:3
5036:N
5031:2
5029:N
5024:1
5022:N
5017:4
5015:T
5010:3
5008:T
5003:2
5001:T
4996:1
4994:T
4970:P
4966:I
4955:M
4951:K
4947:L
4943:J
4939:M
4935:L
4931:K
4927:J
4911:2
4909:M
4905:I
4900:1
4898:M
4893:3
4891:M
4882:3
4880:M
4876:K
4872:J
4868:I
4847:2
4845:M
4840:1
4838:M
4826:3
4823:M
4819:2
4816:M
4812:1
4809:M
4805:I
4771:M
4768:D
4763:M
4760:B
4732:Y
4729:D
4724:W
4721:B
4707:M
4691:Y
4683:W
4630:.
4592:.
4586:c
4583:a
4578:d
4575:b
4569:=
4562:2
4558:l
4552:2
4548:k
4532:e
4528:h
4524:d
4520:g
4516:f
4512:b
4508:l
4504:h
4500:g
4496:c
4492:f
4488:e
4484:a
4480:k
4459:)
4456:h
4453:+
4450:f
4447:(
4444:)
4441:g
4438:+
4435:e
4432:(
4429:)
4426:g
4423:+
4420:f
4417:(
4414:)
4411:h
4408:+
4405:e
4402:(
4397:)
4394:f
4391:e
4388:h
4385:+
4382:e
4379:h
4376:g
4373:+
4370:h
4367:g
4364:f
4361:+
4358:g
4355:f
4352:e
4349:(
4346:2
4340:=
4337:l
4314:,
4308:)
4305:h
4302:+
4299:f
4296:(
4293:)
4290:g
4287:+
4284:e
4281:(
4278:)
4275:h
4272:+
4269:g
4266:(
4263:)
4260:f
4257:+
4254:e
4251:(
4246:)
4243:f
4240:e
4237:h
4234:+
4231:e
4228:h
4225:g
4222:+
4219:h
4216:g
4213:f
4210:+
4207:g
4204:f
4201:e
4198:(
4195:2
4189:=
4186:k
4164:h
4160:g
4156:f
4152:e
4129:.
4122:)
4117:g
4114:e
4111:4
4108:+
4105:)
4102:h
4099:+
4096:f
4093:(
4090:)
4087:g
4084:+
4081:e
4078:(
4073:(
4065:g
4062:+
4059:e
4054:h
4051:+
4048:f
4040:=
4037:q
4014:,
4007:)
4002:h
3999:f
3996:4
3993:+
3990:)
3987:h
3984:+
3981:f
3978:(
3975:)
3972:g
3969:+
3966:e
3963:(
3958:(
3950:h
3947:+
3944:f
3939:g
3936:+
3933:e
3925:=
3922:p
3896:D
3892:C
3888:B
3884:A
3876:h
3872:g
3868:f
3864:e
3842:.
3835:)
3832:e
3829:+
3826:h
3823:(
3820:)
3817:h
3814:+
3811:g
3808:(
3805:)
3802:g
3799:+
3796:f
3793:(
3790:)
3787:f
3784:+
3781:e
3778:(
3773:)
3770:f
3767:e
3764:h
3761:+
3758:e
3755:h
3752:g
3749:+
3746:h
3743:g
3740:f
3737:+
3734:g
3731:f
3728:e
3725:(
3722:)
3719:h
3716:+
3713:g
3710:+
3707:f
3704:+
3701:e
3698:(
3691:=
3667:l
3663:k
3643:.
3636:)
3633:g
3630:+
3627:h
3624:(
3621:)
3618:f
3615:+
3612:h
3609:(
3606:)
3603:e
3600:+
3597:h
3594:(
3589:f
3586:e
3583:h
3580:+
3577:e
3574:h
3571:g
3568:+
3565:h
3562:g
3559:f
3556:+
3553:g
3550:f
3547:e
3540:=
3535:2
3532:D
3503:,
3496:)
3493:h
3490:+
3487:g
3484:(
3481:)
3478:f
3475:+
3472:g
3469:(
3466:)
3463:e
3460:+
3457:g
3454:(
3449:f
3446:e
3443:h
3440:+
3437:e
3434:h
3431:g
3428:+
3425:h
3422:g
3419:f
3416:+
3413:g
3410:f
3407:e
3400:=
3395:2
3392:C
3363:,
3356:)
3353:h
3350:+
3347:f
3344:(
3341:)
3338:g
3335:+
3332:f
3329:(
3326:)
3323:e
3320:+
3317:f
3314:(
3309:f
3306:e
3303:h
3300:+
3297:e
3294:h
3291:g
3288:+
3285:h
3282:g
3279:f
3276:+
3273:g
3270:f
3267:e
3260:=
3255:2
3252:B
3223:,
3216:)
3213:h
3210:+
3207:e
3204:(
3201:)
3198:g
3195:+
3192:e
3189:(
3186:)
3183:f
3180:+
3177:e
3174:(
3169:f
3166:e
3163:h
3160:+
3157:e
3154:h
3151:g
3148:+
3145:h
3142:g
3139:f
3136:+
3133:g
3130:f
3127:e
3120:=
3115:2
3112:A
3083:D
3079:C
3075:B
3071:A
3063:h
3059:g
3055:f
3051:e
3027:2
3023:r
3017:1
3013:r
3007:4
3003:r
2999:+
2994:1
2990:r
2984:4
2980:r
2974:3
2970:r
2966:+
2961:4
2957:r
2951:3
2947:r
2941:2
2937:r
2933:+
2928:3
2924:r
2918:2
2914:r
2908:1
2904:r
2900:=
2897:G
2871:)
2866:4
2862:r
2856:2
2852:r
2848:+
2843:3
2839:r
2833:1
2829:r
2825:(
2822:2
2815:)
2810:4
2806:r
2800:2
2796:r
2792:+
2787:3
2783:r
2777:1
2773:r
2769:(
2764:4
2760:r
2754:3
2750:r
2744:2
2740:r
2734:1
2730:r
2726:4
2718:2
2714:G
2708:+
2705:G
2699:=
2696:r
2667:4
2663:r
2659:,
2654:3
2650:r
2646:,
2641:2
2637:r
2633:,
2628:1
2624:r
2584:)
2581:v
2578:u
2575:y
2572:+
2569:u
2566:y
2563:x
2560:+
2557:y
2554:x
2551:v
2548:+
2545:x
2542:v
2539:u
2536:(
2530:2
2527:1
2521:=
2491:)
2488:x
2485:v
2482:+
2479:y
2476:u
2473:(
2470:)
2467:y
2464:v
2461:+
2458:x
2455:u
2452:(
2449:)
2446:y
2443:x
2440:+
2437:v
2434:u
2431:(
2428:y
2425:x
2422:v
2419:u
2414:)
2411:v
2408:u
2405:y
2396:(
2393:)
2390:u
2387:y
2384:x
2375:(
2372:)
2369:y
2366:x
2363:v
2354:(
2351:)
2348:x
2345:v
2342:u
2333:(
2326:2
2323:=
2320:r
2287:I
2269:.
2262:h
2259:+
2256:g
2253:+
2250:f
2247:+
2244:e
2239:f
2236:e
2233:h
2230:+
2227:e
2224:h
2221:g
2218:+
2215:h
2212:g
2209:f
2206:+
2203:g
2200:f
2197:e
2190:=
2187:r
2154:s
2150:K
2130:d
2127:+
2124:b
2120:K
2115:=
2109:c
2106:+
2103:a
2099:K
2094:=
2089:s
2086:K
2081:=
2078:r
2065:d
2061:c
2057:b
2053:a
2015:r
1984:2
1980:r
1976:4
1970:K
1950:K
1942:r
1925:r
1922:4
1916:s
1903:s
1877:d
1874:c
1871:b
1868:a
1860:K
1847:d
1843:c
1839:b
1835:a
1823:θ
1819:θ
1802:,
1798:|
1783:)
1780:d
1777:b
1771:c
1768:a
1765:(
1761:|
1754:2
1751:1
1745:=
1742:K
1716:.
1711:2
1707:D
1704:+
1701:B
1687:2
1684:D
1671:2
1668:B
1657:b
1654:a
1651:=
1648:K
1632:I
1613:2
1609:C
1606:+
1603:A
1590:)
1586:D
1583:I
1577:B
1574:I
1571:+
1568:C
1565:I
1559:A
1556:I
1552:(
1548:=
1545:K
1505:d
1502:c
1499:b
1496:a
1491:=
1488:K
1452:.
1447:2
1443:D
1440:+
1437:B
1423:d
1420:c
1417:b
1414:a
1409:=
1404:2
1400:C
1397:+
1394:A
1380:d
1377:c
1374:b
1371:a
1366:=
1363:K
1349:d
1345:c
1341:b
1337:a
1309:d
1306:c
1303:b
1300:a
1267:.
1260:2
1256:)
1252:h
1249:f
1243:g
1240:e
1237:(
1231:d
1228:c
1225:b
1222:a
1217:=
1214:K
1179:.
1174:)
1171:f
1168:e
1165:h
1162:+
1159:e
1156:h
1153:g
1150:+
1147:h
1144:g
1141:f
1138:+
1135:g
1132:f
1129:e
1126:(
1123:)
1120:h
1117:+
1114:g
1111:+
1108:f
1105:+
1102:e
1099:(
1094:=
1091:K
1077:h
1073:g
1069:f
1065:e
1054:d
1050:c
1046:b
1042:a
1038:q
1034:p
1012:2
1008:)
1004:d
1001:b
995:c
992:a
989:(
981:2
977:q
971:2
967:p
958:2
955:1
949:=
946:K
928:r
920:s
902:,
899:s
893:r
890:=
887:K
873:K
849:l
845:k
843:(
826:h
822:g
818:f
814:e
812:(
768:d
764:c
760:b
756:a
751:d
747:R
742:c
738:R
733:b
729:R
724:a
720:R
701:d
697:R
691:b
687:R
683:=
678:c
674:R
668:a
664:R
650:d
646:c
642:b
638:a
613:.
608:2
604:C
601:B
598:D
578:2
574:B
571:D
568:A
553:=
548:2
544:C
541:D
538:B
518:2
514:D
511:B
508:A
435::
432:C
429:F
423:F
420:A
417:=
414:C
411:E
405:E
402:A
373:F
370:D
367:+
364:E
361:D
358:=
355:F
352:B
349:+
346:E
343:B
325:F
321:E
306:d
302:b
298:c
294:a
277:.
274:s
271:=
266:2
262:d
259:+
256:c
253:+
250:b
247:+
244:a
238:=
235:d
232:+
229:b
226:=
223:c
220:+
217:a
204:s
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