3242:
5225:
599:
153:
38:
569:
478:
2540:
1251:
3231:
3888:
3479:
216:
2242:
1566:
1255:
Since barycentric coordinates are all positive for a point in a triangle's interior but at least one is negative for a point in the exterior, and two of the barycentric coordinates are zero for a vertex point, the barycentric coordinates given for the orthocenter show that the orthocenter is in an
1999:
1872:
918:
5598:
906:
1726:
2929:
2807:
2966:
5451:
3322:
733:
115:: one-half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the
473:{\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}}
2200:
4308:
3743:
4973:
584:, exterior to the triangle. This is illustrated in the adjacent diagram: in this obtuse triangle, an altitude dropped perpendicularly from the top vertex, which has an acute angle, intersects the extended horizontal side outside the triangle.
2535:{\displaystyle {\begin{aligned}&r_{a}+r_{b}+r_{c}+r={\overline {AH}}+{\overline {BH}}+{\overline {CH}}+2R,\\&r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}={\overline {AH}}^{2}+{\overline {BH}}^{2}+{\overline {CH}}^{2}+(2R)^{2}.\end{aligned}}}
4828:
1405:
4557:
6372:
743:
5079:
5188:
4137:
1885:
1758:
5469:
4678:
4082:
1606:
99:
of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the foot and the apex. The process of drawing the altitude from a vertex to the foot is known as
5474:
2825:
2703:
1246:{\displaystyle {\begin{aligned}&(a^{2}+b^{2}-c^{2})(a^{2}-b^{2}+c^{2}):(a^{2}+b^{2}-c^{2})(-a^{2}+b^{2}+c^{2}):(a^{2}-b^{2}+c^{2})(-a^{2}+b^{2}+c^{2})\\&=\tan A:\tan B:\tan C.\end{aligned}}}
3226:{\displaystyle {\begin{aligned}{\overline {OH}}^{2}&=R^{2}-8R^{2}\cos A\cos B\cos C\\&=9R^{2}-(a^{2}+b^{2}+c^{2}),\\{\overline {HI}}^{2}&=2r^{2}-4R^{2}\cos A\cos B\cos C.\end{aligned}}}
2971:
2830:
2708:
2247:
923:
748:
221:
5324:
2625:
5698:(1810): Draw a line parallel to each side of the triangle through the opposite point, and form a new triangle from the intersections of these three lines. Then the original triangle is the
3709:
3656:
3603:
3978:
1743:
The sum of the ratios on the three altitudes of the distance of the orthocenter from the base to the length of the altitude is 1: (This property and the next one are applications of a
4395:
2697:
of the Euler line, between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half of that between the centroid and the orthocenter:
1375:
628:
572:
In a right triangle, the altitude from each acute angle coincides with a leg and intersects the opposite side at (has its foot at) the right-angled vertex, which is the orthocenter.
3474:{\displaystyle {\begin{array}{rccccc}D=&0&:&\sec B&:&\sec C\\E=&\sec A&:&0&:&\sec C\\F=&\sec A&:&\sec B&:&0\end{array}}}
580:), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite
6314:
550:
2058:
4221:
4861:
5111:
5314:
5277:
1561:{\displaystyle {\vec {OH}}=\sum \limits _{\scriptstyle {\rm {cyclic}}}{\vec {OA}},\qquad 2\cdot {\vec {HO}}=\sum \limits _{\scriptstyle {\rm {cyclic}}}{\vec {HA}}.}
4686:
156:
The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using
3883:{\displaystyle {\begin{array}{rrcrcr}A''=&-a&:&b&:&c\\B''=&a&:&-b&:&c\\C''=&a&:&b&:&-c\end{array}}}
4461:
4986:
5792:
5119:
5622:(3rd century BC), citing the "commentary to the treatise about right-angled triangles", a work which does not survive. It was also mentioned by
4094:
3898:
6707:
1994:{\displaystyle {\frac {\overline {AH}}{\overline {AD}}}+{\frac {\overline {BH}}{\overline {BE}}}+{\frac {\overline {CH}}{\overline {CF}}}=2.}
1867:{\displaystyle {\frac {\overline {HD}}{\overline {AD}}}+{\frac {\overline {HE}}{\overline {BE}}}+{\frac {\overline {HF}}{\overline {CF}}}=1.}
5593:{\displaystyle {\begin{aligned}{\tfrac {1}{2}}AC\cdot BC&={\tfrac {1}{2}}AB\cdot CD\\CD&={\tfrac {AC\cdot BC}{AB}}\\\end{aligned}}}
901:{\displaystyle {\begin{aligned}&\sec A:\sec B:\sec C\\&=\cos A-\sin B\sin C:\cos B-\sin C\sin A:\cos C-\sin A\sin B,\end{aligned}}}
5647:
5812:
Dörrie, Heinrich, "100 Great
Problems of Elementary Mathematics. Their History and Solution". Dover Publications, Inc., New York, 1965,
3502:, posed in 1775. The sides of the orthic triangle are parallel to the tangents to the circumcircle at the original triangle's vertices.
4589:
5767:
3990:
2638:
that are externally tangent to one side of a triangle and tangent to the extensions of the other sides pass through the orthocenter.
1721:{\displaystyle {\overline {AH}}\cdot {\overline {HD}}={\overline {BH}}\cdot {\overline {HE}}={\overline {CH}}\cdot {\overline {HF}}.}
2924:{\displaystyle {\begin{aligned}{\overline {HI}}&<{\overline {HG}},\\{\overline {HG}}&>{\overline {IG}}.\end{aligned}}}
2236:
again as the radius of its circumcircle, the following relations hold regarding the distances of the orthocenter from the vertices:
1878:
The sum of the ratios on the three altitudes of the distance of the orthocenter from the vertex to the length of the altitude is 2:
576:
For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an
2802:{\displaystyle {\begin{aligned}&{\overline {OH}}=2{\overline {NH}},\\&2{\overline {OG}}={\overline {GH}}.\end{aligned}}}
6623:
6059:
5800:
1599:
The product of the lengths of the segments that the orthocenter divides an altitude into is the same for all three altitudes:
911:
5446:{\displaystyle {\frac {1}{h_{c}^{2}}}={\frac {1}{h_{a}^{2}}}+{\frac {1}{h_{b}^{2}}}={\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}.}
2031:
Four points in the plane, such that one of them is the orthocenter of the triangle formed by the other three, is called an
5660:, but was not widely known in Europe, and the theorem was therefore proven several more times in the 17thâ19th century.
2577:
6641:
6583:
6158:
5817:
618:
the triangle is acute. If one angle is a right angle, the orthocenter coincides with the vertex at the right angle.
5608:
The theorem that the three altitudes of a triangle concur (at the orthocenter) is not directly stated in surviving
5845:
17:
6691:
5884:
3924:
728:{\displaystyle a=\left|{\overline {BC}}\right|,b=\left|{\overline {CA}}\right|,c=\left|{\overline {AB}}\right|}
4347:
3327:
1314:
2025:
6010:
Marie-Nicole Gras, "Distances between the circumcenter of the extouch triangle and the classical centers",
5635:
6257:
3748:
3660:
3607:
3554:
5457:
2195:{\displaystyle a^{2}+b^{2}+c^{2}+{\overline {AH}}^{2}+{\overline {BH}}^{2}+{\overline {CH}}^{2}=12R^{2}.}
561:
515:
5684:
3277:
42:
5691:
5212:, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. This is
4303:{\displaystyle \displaystyle {\frac {1}{r}}={\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}.}
3726:
3515:
are parallel to the sides of the orthic triangle, forming a triangle similar to the orthic triangle.
3498:, the inscribed triangle with the smallest perimeter is the orthic triangle. This is the solution to
2631:
4968:{\displaystyle {\overline {AC}}^{2}+{\overline {EB}}^{2}={\overline {AB}}^{2}+{\overline {CE}}^{2}.}
112:
2819:
than it is to the centroid, and the orthocenter is farther than the incenter is from the centroid:
6562:
6484:
6451:
6351:
1735:
116:
134:
of that side as its foot. Also the altitude having the incongruent side as its base will be the
6433:
6421:
6172:
Dorin
Andrica and Dan S ̧tefan Marinescu. "New Interpolation Inequalities to Euler's R ℠2r".
5703:
1576:
553:
157:
6511:
6047:
5656:
This proof in Arabic was translated as part of the (early 17th century) Latin editions of the
5084:
6225:
6015:
5888:
5286:
5249:
3316:
2646:
736:
127:
105:
6350:
4823:{\displaystyle \mathrm {Area} ^{-1}=4{\sqrt {H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}}.}
6523:
6499:
5695:
5623:
5213:
5209:
3891:
3730:
3499:
3487:
of the orthic triangle meet the opposite extended sides of its reference triangle at three
1298:
5771:
8:
6592:
6456:
6400:"Two beautiful geometrical theorems by Abƫ Sahl Kƫhī in a 17th century Dutch translation"
3519:
2032:
593:
31:
6596:
6177:
4552:{\displaystyle {\frac {p_{1}}{h_{1}}}+{\frac {p_{2}}{h_{2}}}+{\frac {p_{3}}{h_{3}}}=1.}
4088:
2017:
2006:
123:
6675:
6460:
6425:
5947:
5913:
3292:. That is, the feet of the altitudes of an oblique triangle form the orthic triangle,
95:
of the altitude. The intersection of the extended base and the altitude is called the
6687:
6658:
6637:
6619:
6579:
6154:
6055:
5975:
5813:
5796:
5720:
5706:
of the new triangle, and therefore concur (at the circumcenter of the new triangle).
5609:
2686:
2658:
66:
6241:
Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle",
6212:
Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle,"
5963:
3241:
6681:
6539:
6442:
6387:
5643:
3488:
1396:
5948:
Bell, Amy, "Hansen's right triangle theorem, its converse and a generalization",
5715:
5699:
5661:
5074:{\displaystyle {\tfrac {1}{2}}ah_{a}={\tfrac {1}{2}}bh_{b}={\tfrac {1}{2}}ch_{c}}
3299:. Also, the incenter (the center of the inscribed circle) of the orthic triangle
1265:
85:
80:
71:
5974:
Weisstein, Eric W. "Jerabek
Hyperbola." From MathWorld--A Wolfram Web Resource.
5924:
5912:
Weisstein, Eric W. "Isotomic conjugate" From MathWorld--A Wolfram Web
Resource.
1744:
606:
The three (possibly extended) altitudes intersect in a single point, called the
30:"Orthocenter" and "Orthocentre" redirect here. For the orthocentric system, see
5962:
Weisstein, Eric W. "Kiepert
Parabola." From MathWorld--A Wolfram Web Resource.
5614:
3740:
Trilinear coordinates for the vertices of the tangential triangle are given by
3495:
3281:
1280:
1261:
1257:
615:
484:
135:
6661:
6544:
6527:
6446:
6391:
5224:
6701:
3919:
3729:
to the orthic triangle. The circumcenter of the tangential triangle, and the
3484:
2021:
1272:
581:
91:
76:
41:
The three altitudes of a triangle intersect at the orthocenter, which for an
3508:
The tangent lines of the nine-point circle at the midpoints of the sides of
6346:
5669:
5183:{\displaystyle {\frac {1}{h_{a}}}<{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}}
4334:
2954:
2675:
2045:
2010:
577:
130:
sides), the altitude having the incongruent side as its base will have the
62:
5228:
Comparison of the inverse
Pythagorean theorem with the Pythagorean theorem
4132:{\displaystyle {\tfrac {1}{2}}\times {\text{base}}\times {\text{height}},}
5925:
Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Web
Resource.
3505:
The orthic triangle of an acute triangle gives a triangular light route.
2642:
149:), often subscripted with the name of the side the altitude is drawn to.
5702:
of the new triangle, and the altitudes of the original triangle are the
5849:
5619:
4091:
for the area of a triangle in terms of the sides with the area formula
3734:
2690:
598:
488:
152:
37:
5885:
Panapoi, Ronnachai, "Some properties of the orthocenter of a triangle"
6666:
6399:
27:
Perpendicular line segment from a triangle's side to opposite vertex
4586:, and denoting the semi-sum of the reciprocals of the altitudes as
4208:
2947:
2813:
2694:
2668:
2652:
2635:
2229:
2210:
131:
89:). This (infinite) line containing the (finite) base is called the
58:
50:
6489:
Solutions peu connues de différens problÚmes de Géométrie-pratique
6029:
Smith, Geoff, and
Leversha, Gerry, "Euler and triangle geometry",
1734:
having radius the square root of this constant is the triangle's
1571:
The first of the previous vector identities is also known as the
625:
denote the vertices and also the angles of the triangle, and let
4673:{\displaystyle H={\tfrac {h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1}}{2}}}
1391:. From this, the following characterizations of the orthocenter
1308:
is located at the origin of the plane. Then, the complex number
4077:{\displaystyle h_{a}={\frac {2{\sqrt {s(s-a)(s-b)(s-c)}}}{a}}.}
1748:
6045:
5795:, "Complex numbers from A to...Z". BirkhÀuser, Boston, 2006,
6493:
Little-known solutions of various
Geometry practice problems
6359:. Vol. 4. Cambridge University Press. pp. 454â455.
6329:
Richinick, Jennifer, "The upside-down
Pythagorean Theorem,"
568:
6528:"The Triangle and its Six Scribed Circles §5. Orthocentre"
6518:. Vol. 4. Göttingen Academy of Sciences. p. 396.
6506:. By Carnot, Lazare (in German). Translated by Schumacher.
6404:
TÄrÄ«kÍh-e ÊŸElm: Iranian Journal for the History of Science
6016:
http://forumgeom.fau.edu/FG2014volume14/FG201405index.html
3284:
of the orthocenter of the original triangle is called the
6495:] (in French). Devilly, Metz et Courcier. p. 15.
6461:"A Possibly First Proof of the Concurrence of Altitudes"
6135:
Bryant, V., and Bradley, H., "Triangular Light Routes,"
5634:
340). The theorem was stated and proved explicitly by
5556:
5512:
5478:
5047:
5019:
4991:
4600:
4099:
3935:
1512:
1434:
6260:
6232:, Dover Publishing Co., second revised edition, 1996.
5472:
5327:
5289:
5252:
5122:
5087:
4989:
4864:
4689:
4592:
4464:
4350:
4318:
Denoting the altitude from one side of a triangle as
4225:
4224:
4097:
3993:
3927:
3746:
3663:
3610:
3557:
3325:
3319:
for the vertices of the orthic triangle are given by
2969:
2828:
2706:
2580:
2245:
2061:
1888:
1761:
1609:
1408:
1317:
921:
746:
631:
518:
219:
6178:
http://forumgeom.fau.edu/FG2017volume17/FG201719.pdf
5765:
Clark Kimberling's Encyclopedia of Triangle Centers
4166:, this equation can also used to find the altitudes
3904:
3529:
be the line tangent to the circumcircle of triangle
494:
divides the hypotenuse into two segments of lengths
5914:
http://mathworld.wolfram.com/IsotomicConjugate.html
4337:(radius of the triangle's circumscribed circle) as
3890:The reference triangle and its orthic triangle are
3733:of the orthic and tangential triangles, are on the
2645:passing through the orthocenter of a triangle is a
2039:
6682:Animated demonstration of orthocenter construction
6308:
5976:http://mathworld.wolfram.com/JerabekHyperbola.html
5592:
5445:
5318:. The third altitude can be found by the relation
5308:
5271:
5182:
5105:
5073:
4967:
4822:
4672:
4567:Denoting the altitudes of any triangle from sides
4551:
4389:
4302:
4131:
4076:
3972:
3882:
3703:
3650:
3597:
3473:
3225:
2923:
2801:
2693:. The center of the nine-point circle lies at the
2620:{\displaystyle {\overline {HD}}={\overline {DP}}.}
2619:
2534:
2194:
1993:
1866:
1720:
1560:
1369:
1245:
900:
727:
544:
472:
141:It is common to mark the altitude with the letter
6656:
6532:Proceedings of the Edinburgh Mathematical Society
5964:http://mathworld.wolfram.com/KiepertParabola.html
3909:
3897:For more information on the orthic triangle, see
455:
451:
390:
386:
385:
352:
348:
347:
316:
312:
302:
298:
297:
6699:
6561:
6087:
5935:
4455:are the altitudes to the respective sides, then
3259:in the text) is the orthic triangle of triangle
2653:Relation to other centers, the nine-point circle
111:Altitudes can be used in the computation of the
4833:
4428:are the perpendicular distances from any point
4143:and the height is the altitude from the vertex
2551:, is extended to intersect the circumcircle at
602:Three altitudes intersecting at the orthocenter
6361:Note Whiteside's footnotes 90â92, pp. 454â456.
6054:. American Mathematical Society. p. 292.
3518:The orthic triangle is closely related to the
2561:is a chord of the circumcircle, then the foot
6573:
6111:
5998:
5986:
5753:
502:. If we denote the length of the altitude by
6373:"Concurrency of the Altitudes of a Triangle"
6370:
3306:is the orthocenter of the original triangle
614:. The orthocenter lies inside the triangle
6219:
3718:, whose sides are the tangents to triangle
4188:Consider an arbitrary triangle with sides
1260:interior, on the right-angled vertex of a
735:be the side lengths. The orthocenter has
444:
423:
422:
421:
267:
266:
247:
246:
6543:
6455:
6397:
6341:
6339:
6052:Continuous symmetry: from Euclid to Klein
6041:
6039:
5880:
5878:
5840:
5838:
3973:{\displaystyle s={\tfrac {1}{2}}(a+b+c),}
268:
6371:Hajja, Mowaffaq; Martini, Horst (2013).
5785:
5638:in his (11th century) commentary on the
5244:, each of the legs is also an altitude:
5223:
3240:
597:
567:
151:
104:at that vertex. It is a special case of
36:
6613:
6483:
6357:The Mathematical Papers of Isaac Newton
6200:
6188:
6151:College Geometry / A Discovery Approach
6123:
6099:
6093:
6075:
5869:
5829:
5749:
5747:
5690:A particularly elegant proof is due to
5199:
4390:{\displaystyle h_{a}={\frac {bc}{2R}}.}
4313:
3725:'s circumcircle at its vertices; it is
2689:all lie on a single line, known as the
1370:{\displaystyle z_{H}=z_{A}+z_{B}+z_{C}}
14:
6700:
6574:Berele, Allan; Goldman, Jerry (2001),
6522:
6420:
6345:
6336:
6046:William H. Barker, Roger Howe (2007).
6036:
5875:
5835:
4978:
3280:(does not contain a right-angle), the
1591:denote the feet of the altitudes from
1399:can be established straightforwardly:
6708:Straight lines defined for a triangle
6657:
6631:
6591:
6509:
6498:
6025:
6023:
5900:
5806:
5738:
6576:Geometry: Theorems and Constructions
6355:. In Whiteside, Derek Thomas (ed.).
6352:"3.1 The 'Geometry of Curved Lines'"
6309:{\displaystyle a^{-2}+b^{-2}=d^{-2}}
6254:Voles, Roger, "Integer solutions of
6048:"§ VI.2: The classical coincidences"
5744:
5672:proved it in an unfinished treatise
4183:
3704:{\displaystyle C''=L_{C}\cap L_{A}.}
3651:{\displaystyle B''=L_{C}\cap L_{A},}
3598:{\displaystyle A''=L_{B}\cap L_{C},}
1747:of any interior point and the three
610:of the triangle, usually denoted by
6176:, Volume 17 (2017), pp. 149â156.
6148:
1508:
1430:
557:
24:
6020:
5768:"Encyclopedia of Triangle Centers"
4983:Since the area of the triangle is
4701:
4698:
4695:
4692:
3236:
1530:
1527:
1524:
1521:
1518:
1515:
1452:
1449:
1446:
1443:
1440:
1437:
1384:, namely the altitude of triangle
545:{\displaystyle h_{c}={\sqrt {pq}}}
25:
6719:
6650:
5759:
5219:
4400:
4192:and with corresponding altitudes
3905:Some additional altitude theorems
2812:The orthocenter is closer to the
79:to a line containing the side or
6230:Challenging Problems in Geometry
6066:See also: Corollary 5.5, p. 318.
5194:
4139:where the base is taken as side
2209:as the radius of the triangle's
2040:Relation with circles and conics
6477:
6414:
6364:
6323:
6248:
6235:
6206:
6194:
6182:
6166:
6142:
6129:
6117:
6105:
6081:
6069:
6004:
5992:
5980:
5968:
5956:
5941:
5929:
5918:
5906:
5894:
5618:(proposition 5), attributed to
4562:
3522:, constructed as follows: let
1480:
420:
272:
255:
6692:Wolfram Demonstrations Project
6597:"Existence of the Orthocenter"
6510:Gauss, Carl Friedrich (1873).
6426:"XXIV. Geometry and geometers"
6380:Mathematische Semesterberichte
5863:
5823:
5732:
5232:In a right triangle with legs
4812:
4785:
4782:
4755:
4752:
4725:
4060:
4048:
4045:
4033:
4030:
4018:
3964:
3946:
3910:Altitude in terms of the sides
3121:
3082:
2545:If any altitude, for example,
2516:
2506:
1549:
1498:
1471:
1420:
1190:
1148:
1145:
1106:
1100:
1058:
1055:
1016:
1010:
971:
968:
929:
587:
237:
224:
13:
1:
6636:(5th ed.), Brooks/Cole,
6555:
5846:""Orthocenter of a triangle""
5678:
5631:
1582:
6153:, HarperCollins, p. 6,
4951:
4926:
4901:
4876:
4842:is any point on an altitude
4834:General point on an altitude
4087:This follows from combining
3914:For any triangle with sides
3711:The tangential triangle is
3142:
2985:
2909:
2887:
2865:
2843:
2787:
2769:
2743:
2722:
2609:
2591:
2492:
2467:
2442:
2342:
2324:
2306:
2162:
2137:
2112:
1979:
1966:
1946:
1933:
1913:
1900:
1852:
1839:
1819:
1806:
1786:
1773:
1710:
1692:
1674:
1656:
1638:
1620:
1380:is represented by the point
716:
684:
652:
509:, we then have the relation
487:, the altitude drawn to the
160:on the 3 triangles of sides
7:
6616:Advanced Euclidean Geometry
6614:Johnson, Roger A. (2007) ,
6033:91, November 2007, 436â452.
5709:
5458:inverse Pythagorean theorem
4341:, the altitude is given by
2035:or orthocentric quadrangle.
562:inverse Pythagorean theorem
10:
6724:
6678:With interactive animation
6398:Hogendijk, Jan P. (2008).
5612:texts, but is used in the
5603:
5456:This is also known as the
5081:, the triangle inequality
2656:
2026:anticomplementary triangle
2020:of the orthocenter is the
2009:of the orthocenter is the
591:
29:
6684:Compass and straightedge.
6676:Orthocenter of a triangle
6545:10.1017/S0013091500036762
6447:10.1080/14786445008646583
6392:10.1007/s00591-013-0123-z
6112:Berele & Goldman 2001
5999:Berele & Goldman 2001
5987:Berele & Goldman 2001
5754:Berele & Goldman 2001
5694:(1804) and independently
4325:, the other two sides as
6632:Smart, James R. (1998),
6563:Altshiller-Court, Nathan
6485:Servois, Francois-Joseph
6245:89 (November 2005), 494.
6228:and Charles T. Salkind,
5803:, page 90, Proposition 3
5726:
5674:Geometry of Curved Lines
5106:{\displaystyle a<b+c}
4207:. The altitudes and the
6452:Footnote on pp. 207â208
6422:Davies, Thomas Stephens
6333:92, July 2008, 313â317.
6320:83, July 1999, 269â271.
6216:89, November 2005, 494.
6139:82, July 1998, 298-299.
5704:perpendicular bisectors
5692:François-Joseph Servois
5628:Mathematical Collection
5309:{\displaystyle h_{b}=a}
5272:{\displaystyle h_{a}=b}
3984:(the base) is given by
3980:the altitude from side
1730:The circle centered at
912:barycentric coordinates
117:trigonometric functions
83:opposite the apex (the
45:is inside the triangle.
6504:Geometrie der Stellung
6434:Philosophical Magazine
6310:
6149:Kay, David C. (1993),
6126:, p. 172, Section 270c
5594:
5447:
5310:
5273:
5229:
5184:
5107:
5075:
4969:
4824:
4674:
4553:
4391:
4304:
4133:
4078:
3974:
3884:
3705:
3652:
3599:
3475:
3266:
3227:
2934:In terms of the sides
2925:
2803:
2621:
2536:
2205:In addition, denoting
2196:
1995:
1868:
1722:
1577:James Joseph Sylvester
1562:
1371:
1247:
902:
729:
603:
573:
554:Geometric mean theorem
546:
480:
474:
46:
6524:Mackay, John Sturgeon
6500:Gauss, Carl Friedrich
6311:
6226:Alfred S. Posamentier
6203:, p. 74, Section 103c
6191:, p. 71, Section 101a
6102:, p. 168, Section 264
6088:Altshiller-Court 2007
6078:, p. 199, Section 315
5936:Altshiller-Court 2007
5889:University of Georgia
5872:, p. 176, Section 278
5832:, p. 163, Section 255
5595:
5448:
5311:
5274:
5227:
5185:
5108:
5076:
4970:
4825:
4675:
4554:
4392:
4333:, and the triangle's
4305:
4134:
4079:
3975:
3885:
3706:
3653:
3600:
3476:
3317:Trilinear coordinates
3244:
3228:
2926:
2804:
2647:rectangular hyperbola
2622:
2537:
2197:
1996:
1869:
1745:more general property
1723:
1563:
1372:
1264:, and exterior to an
1248:
903:
737:trilinear coordinates
730:
601:
571:
547:
475:
155:
138:of the vertex angle.
126:(a triangle with two
106:orthogonal projection
102:dropping the altitude
40:
6593:Bogomolny, Alexander
6457:Bogomolny, Alexander
6331:Mathematical Gazette
6318:Mathematical Gazette
6258:
6243:Mathematical Gazette
6214:Mathematical Gazette
6137:Mathematical Gazette
6031:Mathematical Gazette
5696:Carl Friedrich Gauss
5642:, and attributed to
5470:
5463:Note in particular:
5325:
5287:
5250:
5210:equilateral triangle
5200:Equilateral triangle
5120:
5085:
4987:
4862:
4687:
4590:
4462:
4348:
4314:Circumradius theorem
4222:
4095:
3991:
3925:
3892:orthologic triangles
3744:
3731:center of similitude
3661:
3608:
3555:
3323:
2967:
2826:
2704:
2578:
2243:
2228:as the radii of its
2059:
1886:
1759:
1607:
1595:respectively. Then:
1573:problem of Sylvester
1406:
1315:
1297:and assume that the
919:
744:
629:
516:
217:
6690:by Jay Warendorff,
6502:(1810). "ZusÀtze".
6174:Forum Geometricorum
6014:14 (2014), 51-61.
6012:Forum Geometricorum
5950:Forum Geometricorum
5687:proved it in 1749.
5651: 10th century
5397:
5372:
5347:
4979:Triangle inequality
4811:
4781:
4751:
4662:
4641:
4620:
3520:tangential triangle
2414:
2396:
2378:
2048:of the triangle by
2033:orthocentric system
594:Orthocentric system
158:Pythagoras' theorem
32:Orthocentric system
6659:Weisstein, Eric W.
6306:
5610:Greek mathematical
5590:
5588:
5584:
5521:
5487:
5443:
5383:
5358:
5333:
5306:
5269:
5230:
5180:
5103:
5071:
5056:
5028:
5000:
4965:
4820:
4794:
4764:
4734:
4670:
4668:
4645:
4624:
4603:
4549:
4432:to the sides, and
4387:
4300:
4299:
4129:
4108:
4074:
3970:
3944:
3880:
3878:
3701:
3648:
3595:
3551:analogously. Let
3471:
3469:
3267:
3223:
3221:
2921:
2919:
2799:
2797:
2617:
2532:
2530:
2400:
2382:
2364:
2192:
2018:isotomic conjugate
2007:isogonal conjugate
1991:
1864:
1718:
1558:
1537:
1535:
1459:
1457:
1367:
1275:, let the points
1243:
1241:
898:
896:
725:
604:
574:
542:
481:
470:
468:
124:isosceles triangle
113:area of a triangle
47:
6688:Fagnano's Problem
6634:Modern Geometries
6625:978-0-486-46237-0
6578:, Prentice Hall,
6061:978-0-8218-3900-3
5952:6, 2006, 335â342.
5801:978-0-8176-4326-3
5791:Andreescu, Titu;
5721:Median (geometry)
5664:proved it in his
5583:
5520:
5486:
5438:
5418:
5398:
5373:
5348:
5214:Viviani's theorem
5178:
5158:
5138:
5055:
5027:
4999:
4954:
4929:
4904:
4879:
4815:
4667:
4541:
4514:
4487:
4382:
4294:
4274:
4254:
4234:
4184:Inradius theorems
4124:
4116:
4107:
4069:
4063:
3943:
3500:Fagnano's problem
3290:altitude triangle
3145:
2988:
2912:
2890:
2868:
2846:
2790:
2772:
2746:
2725:
2687:nine-point circle
2681:, and the center
2659:Nine-point circle
2612:
2594:
2495:
2470:
2445:
2345:
2327:
2309:
2165:
2140:
2115:
1983:
1982:
1969:
1950:
1949:
1936:
1917:
1916:
1903:
1856:
1855:
1842:
1823:
1822:
1809:
1790:
1789:
1776:
1713:
1695:
1677:
1659:
1641:
1623:
1552:
1507:
1501:
1474:
1429:
1423:
719:
687:
655:
540:
464:
405:
367:
16:(Redirected from
6715:
6672:
6671:
6646:
6628:
6610:
6608:
6607:
6588:
6570:
6567:College Geometry
6550:
6549:
6547:
6519:
6507:
6496:
6481:
6475:
6474:
6472:
6471:
6450:
6441:(249): 198â212.
6430:
6418:
6412:
6411:
6395:
6377:
6368:
6362:
6360:
6354:
6343:
6334:
6327:
6321:
6315:
6313:
6312:
6307:
6305:
6304:
6289:
6288:
6273:
6272:
6252:
6246:
6239:
6233:
6223:
6217:
6210:
6204:
6198:
6192:
6186:
6180:
6170:
6164:
6163:
6146:
6140:
6133:
6127:
6121:
6115:
6109:
6103:
6097:
6091:
6085:
6079:
6073:
6067:
6065:
6043:
6034:
6027:
6018:
6008:
6002:
5996:
5990:
5984:
5978:
5972:
5966:
5960:
5954:
5945:
5939:
5933:
5927:
5922:
5916:
5910:
5904:
5898:
5892:
5882:
5873:
5867:
5861:
5860:
5858:
5857:
5848:. Archived from
5842:
5833:
5827:
5821:
5810:
5804:
5789:
5783:
5782:
5780:
5779:
5770:. Archived from
5763:
5757:
5751:
5742:
5736:
5682:
5680:
5652:
5649:
5633:
5599:
5597:
5596:
5591:
5589:
5585:
5582:
5574:
5557:
5522:
5513:
5488:
5479:
5452:
5450:
5449:
5444:
5439:
5437:
5436:
5424:
5419:
5417:
5416:
5404:
5399:
5396:
5391:
5379:
5374:
5371:
5366:
5354:
5349:
5346:
5341:
5329:
5317:
5315:
5313:
5312:
5307:
5299:
5298:
5280:
5278:
5276:
5275:
5270:
5262:
5261:
5243:
5239:
5235:
5207:
5189:
5187:
5186:
5181:
5179:
5177:
5176:
5164:
5159:
5157:
5156:
5144:
5139:
5137:
5136:
5124:
5112:
5110:
5109:
5104:
5080:
5078:
5077:
5072:
5070:
5069:
5057:
5048:
5042:
5041:
5029:
5020:
5014:
5013:
5001:
4992:
4974:
4972:
4971:
4966:
4961:
4960:
4955:
4950:
4942:
4936:
4935:
4930:
4925:
4917:
4911:
4910:
4905:
4900:
4892:
4886:
4885:
4880:
4875:
4867:
4854:
4848:of any triangle
4847:
4846:
4841:
4829:
4827:
4826:
4821:
4816:
4810:
4802:
4780:
4772:
4750:
4742:
4721:
4713:
4712:
4704:
4679:
4677:
4676:
4671:
4669:
4663:
4661:
4653:
4640:
4632:
4619:
4611:
4601:
4585:
4571:respectively as
4570:
4558:
4556:
4555:
4550:
4542:
4540:
4539:
4530:
4529:
4520:
4515:
4513:
4512:
4503:
4502:
4493:
4488:
4486:
4485:
4476:
4475:
4466:
4454:
4431:
4427:
4396:
4394:
4393:
4388:
4383:
4381:
4373:
4365:
4360:
4359:
4340:
4332:
4328:
4324:
4309:
4307:
4306:
4301:
4295:
4293:
4292:
4280:
4275:
4273:
4272:
4260:
4255:
4253:
4252:
4240:
4235:
4227:
4214:
4206:
4191:
4180:, respectively.
4179:
4172:
4165:
4161:
4157:
4150:
4146:
4142:
4138:
4136:
4135:
4130:
4125:
4122:
4117:
4114:
4109:
4100:
4083:
4081:
4080:
4075:
4070:
4065:
4064:
4014:
4008:
4003:
4002:
3983:
3979:
3977:
3976:
3971:
3945:
3936:
3917:
3889:
3887:
3886:
3881:
3879:
3844:
3801:
3758:
3724:
3717:
3710:
3708:
3707:
3702:
3697:
3696:
3684:
3683:
3671:
3657:
3655:
3654:
3649:
3644:
3643:
3631:
3630:
3618:
3604:
3602:
3601:
3596:
3591:
3590:
3578:
3577:
3565:
3550:
3539:
3535:
3528:
3514:
3489:collinear points
3480:
3478:
3477:
3472:
3470:
3312:
3305:
3298:
3275:
3269:If the triangle
3265:
3258:
3251:
3232:
3230:
3229:
3224:
3222:
3188:
3187:
3172:
3171:
3152:
3151:
3146:
3141:
3133:
3120:
3119:
3107:
3106:
3094:
3093:
3078:
3077:
3059:
3028:
3027:
3012:
3011:
2995:
2994:
2989:
2984:
2976:
2959:
2952:
2945:
2941:
2937:
2930:
2928:
2927:
2922:
2920:
2913:
2908:
2900:
2891:
2886:
2878:
2869:
2864:
2856:
2847:
2842:
2834:
2818:
2808:
2806:
2805:
2800:
2798:
2791:
2786:
2778:
2773:
2768:
2760:
2754:
2747:
2742:
2734:
2726:
2721:
2713:
2710:
2684:
2680:
2673:
2666:
2663:The orthocenter
2626:
2624:
2623:
2618:
2613:
2608:
2600:
2595:
2590:
2582:
2570:
2569:
2565:bisects segment
2564:
2560:
2559:
2554:
2550:
2549:
2541:
2539:
2538:
2533:
2531:
2524:
2523:
2502:
2501:
2496:
2491:
2483:
2477:
2476:
2471:
2466:
2458:
2452:
2451:
2446:
2441:
2433:
2427:
2426:
2413:
2408:
2395:
2390:
2377:
2372:
2362:
2346:
2341:
2333:
2328:
2323:
2315:
2310:
2305:
2297:
2286:
2285:
2273:
2272:
2260:
2259:
2249:
2235:
2227:
2208:
2201:
2199:
2198:
2193:
2188:
2187:
2172:
2171:
2166:
2161:
2153:
2147:
2146:
2141:
2136:
2128:
2122:
2121:
2116:
2111:
2103:
2097:
2096:
2084:
2083:
2071:
2070:
2051:
2013:of the triangle.
2000:
1998:
1997:
1992:
1984:
1978:
1970:
1965:
1957:
1956:
1951:
1945:
1937:
1932:
1924:
1923:
1918:
1912:
1904:
1899:
1891:
1890:
1873:
1871:
1870:
1865:
1857:
1851:
1843:
1838:
1830:
1829:
1824:
1818:
1810:
1805:
1797:
1796:
1791:
1785:
1777:
1772:
1764:
1763:
1733:
1727:
1725:
1724:
1719:
1714:
1709:
1701:
1696:
1691:
1683:
1678:
1673:
1665:
1660:
1655:
1647:
1642:
1637:
1629:
1624:
1619:
1611:
1594:
1590:
1567:
1565:
1564:
1559:
1554:
1553:
1548:
1540:
1536:
1534:
1533:
1503:
1502:
1497:
1489:
1476:
1475:
1470:
1462:
1458:
1456:
1455:
1425:
1424:
1419:
1411:
1394:
1390:
1383:
1376:
1374:
1373:
1368:
1366:
1365:
1353:
1352:
1340:
1339:
1327:
1326:
1307:
1296:
1278:
1258:acute triangle's
1252:
1250:
1249:
1244:
1242:
1196:
1189:
1188:
1176:
1175:
1163:
1162:
1144:
1143:
1131:
1130:
1118:
1117:
1099:
1098:
1086:
1085:
1073:
1072:
1054:
1053:
1041:
1040:
1028:
1027:
1009:
1008:
996:
995:
983:
982:
967:
966:
954:
953:
941:
940:
925:
907:
905:
904:
899:
897:
788:
750:
734:
732:
731:
726:
724:
720:
715:
707:
692:
688:
683:
675:
660:
656:
651:
643:
624:
613:
551:
549:
548:
543:
541:
533:
528:
527:
508:
501:
497:
493:
479:
477:
476:
471:
469:
465:
457:
443:
442:
406:
401:
400:
399:
384:
383:
373:
368:
363:
362:
361:
346:
345:
335:
326:
325:
296:
295:
282:
281:
265:
264:
245:
244:
211:
195:
179:
144:
65:through a given
21:
6723:
6722:
6718:
6717:
6716:
6714:
6713:
6712:
6698:
6697:
6653:
6644:
6626:
6605:
6603:
6586:
6558:
6553:
6520:
6508:republished in
6497:
6482:
6478:
6469:
6467:
6428:
6419:
6415:
6396:
6375:
6369:
6365:
6344:
6337:
6328:
6324:
6297:
6293:
6281:
6277:
6265:
6261:
6259:
6256:
6255:
6253:
6249:
6240:
6236:
6224:
6220:
6211:
6207:
6199:
6195:
6187:
6183:
6171:
6167:
6161:
6147:
6143:
6134:
6130:
6122:
6118:
6110:
6106:
6098:
6094:
6086:
6082:
6074:
6070:
6062:
6044:
6037:
6028:
6021:
6009:
6005:
5997:
5993:
5985:
5981:
5973:
5969:
5961:
5957:
5946:
5942:
5934:
5930:
5923:
5919:
5911:
5907:
5899:
5895:
5883:
5876:
5868:
5864:
5855:
5853:
5844:
5843:
5836:
5828:
5824:
5811:
5807:
5790:
5786:
5777:
5775:
5766:
5764:
5760:
5752:
5745:
5737:
5733:
5729:
5716:Triangle center
5712:
5700:medial triangle
5685:William Chapple
5676:
5662:Samuel Marolois
5650:
5606:
5587:
5586:
5575:
5558:
5555:
5548:
5539:
5538:
5511:
5504:
5477:
5473:
5471:
5468:
5467:
5432:
5428:
5423:
5412:
5408:
5403:
5392:
5387:
5378:
5367:
5362:
5353:
5342:
5337:
5328:
5326:
5323:
5322:
5294:
5290:
5288:
5285:
5284:
5282:
5257:
5253:
5251:
5248:
5247:
5245:
5241:
5240:and hypotenuse
5237:
5233:
5222:
5205:
5204:From any point
5202:
5197:
5172:
5168:
5163:
5152:
5148:
5143:
5132:
5128:
5123:
5121:
5118:
5117:
5086:
5083:
5082:
5065:
5061:
5046:
5037:
5033:
5018:
5009:
5005:
4990:
4988:
4985:
4984:
4981:
4956:
4943:
4941:
4940:
4931:
4918:
4916:
4915:
4906:
4893:
4891:
4890:
4881:
4868:
4866:
4865:
4863:
4860:
4859:
4849:
4844:
4843:
4839:
4836:
4803:
4798:
4773:
4768:
4743:
4738:
4720:
4705:
4691:
4690:
4688:
4685:
4684:
4654:
4649:
4633:
4628:
4612:
4607:
4602:
4599:
4591:
4588:
4587:
4584:
4580:
4576:
4572:
4568:
4565:
4535:
4531:
4525:
4521:
4519:
4508:
4504:
4498:
4494:
4492:
4481:
4477:
4471:
4467:
4465:
4463:
4460:
4459:
4453:
4446:
4439:
4433:
4429:
4426:
4419:
4412:
4406:
4403:
4374:
4366:
4364:
4355:
4351:
4349:
4346:
4345:
4338:
4330:
4326:
4323:
4319:
4316:
4288:
4284:
4279:
4268:
4264:
4259:
4248:
4244:
4239:
4226:
4223:
4220:
4219:
4215:are related by
4212:
4205:
4201:
4197:
4193:
4189:
4186:
4178:
4174:
4171:
4167:
4163:
4159:
4155:
4148:
4147:(opposite side
4144:
4140:
4121:
4113:
4098:
4096:
4093:
4092:
4089:Heron's formula
4013:
4009:
4007:
3998:
3994:
3992:
3989:
3988:
3981:
3934:
3926:
3923:
3922:
3915:
3912:
3907:
3877:
3876:
3868:
3863:
3858:
3853:
3848:
3837:
3834:
3833:
3828:
3823:
3815:
3810:
3805:
3794:
3791:
3790:
3785:
3780:
3775:
3770:
3762:
3751:
3747:
3745:
3742:
3741:
3719:
3712:
3692:
3688:
3679:
3675:
3664:
3662:
3659:
3658:
3639:
3635:
3626:
3622:
3611:
3609:
3606:
3605:
3586:
3582:
3573:
3569:
3558:
3556:
3553:
3552:
3549:
3545:
3541:
3537:
3530:
3527:
3523:
3509:
3468:
3467:
3462:
3457:
3446:
3441:
3430:
3421:
3420:
3409:
3404:
3399:
3394:
3383:
3374:
3373:
3362:
3357:
3346:
3341:
3336:
3326:
3324:
3321:
3320:
3307:
3300:
3293:
3286:orthic triangle
3270:
3260:
3253:
3252:(respectively,
3246:
3239:
3237:Orthic triangle
3220:
3219:
3183:
3179:
3167:
3163:
3153:
3147:
3134:
3132:
3131:
3128:
3127:
3115:
3111:
3102:
3098:
3089:
3085:
3073:
3069:
3057:
3056:
3023:
3019:
3007:
3003:
2996:
2990:
2977:
2975:
2974:
2970:
2968:
2965:
2964:
2957:
2950:
2943:
2939:
2935:
2918:
2917:
2901:
2899:
2892:
2879:
2877:
2874:
2873:
2857:
2855:
2848:
2835:
2833:
2829:
2827:
2824:
2823:
2816:
2796:
2795:
2779:
2777:
2761:
2759:
2752:
2751:
2735:
2733:
2714:
2712:
2707:
2705:
2702:
2701:
2682:
2678:
2671:
2664:
2661:
2655:
2601:
2599:
2583:
2581:
2579:
2576:
2575:
2567:
2566:
2562:
2557:
2556:
2552:
2547:
2546:
2529:
2528:
2519:
2515:
2497:
2484:
2482:
2481:
2472:
2459:
2457:
2456:
2447:
2434:
2432:
2431:
2422:
2418:
2409:
2404:
2391:
2386:
2373:
2368:
2360:
2359:
2334:
2332:
2316:
2314:
2298:
2296:
2281:
2277:
2268:
2264:
2255:
2251:
2246:
2244:
2241:
2240:
2233:
2226:
2222:
2218:
2214:
2206:
2183:
2179:
2167:
2154:
2152:
2151:
2142:
2129:
2127:
2126:
2117:
2104:
2102:
2101:
2092:
2088:
2079:
2075:
2066:
2062:
2060:
2057:
2056:
2049:
2042:
2022:symmedian point
1971:
1958:
1955:
1938:
1925:
1922:
1905:
1892:
1889:
1887:
1884:
1883:
1844:
1831:
1828:
1811:
1798:
1795:
1778:
1765:
1762:
1760:
1757:
1756:
1731:
1702:
1700:
1684:
1682:
1666:
1664:
1648:
1646:
1630:
1628:
1612:
1610:
1608:
1605:
1604:
1592:
1588:
1585:
1541:
1539:
1538:
1514:
1513:
1511:
1490:
1488:
1487:
1463:
1461:
1460:
1436:
1435:
1433:
1412:
1410:
1409:
1407:
1404:
1403:
1392:
1385:
1381:
1361:
1357:
1348:
1344:
1335:
1331:
1322:
1318:
1316:
1313:
1312:
1302:
1295:
1291:
1287:
1283:
1276:
1266:obtuse triangle
1240:
1239:
1194:
1193:
1184:
1180:
1171:
1167:
1158:
1154:
1139:
1135:
1126:
1122:
1113:
1109:
1094:
1090:
1081:
1077:
1068:
1064:
1049:
1045:
1036:
1032:
1023:
1019:
1004:
1000:
991:
987:
978:
974:
962:
958:
949:
945:
936:
932:
922:
920:
917:
916:
895:
894:
786:
785:
747:
745:
742:
741:
708:
706:
702:
676:
674:
670:
644:
642:
638:
630:
627:
626:
622:
611:
596:
590:
532:
523:
519:
517:
514:
513:
507:
503:
499:
495:
491:
467:
466:
456:
438:
434:
424:
408:
407:
395:
391:
379:
375:
374:
372:
357:
353:
341:
337:
336:
334:
327:
321:
317:
291:
287:
284:
283:
277:
273:
260:
256:
248:
240:
236:
220:
218:
215:
214:
197:
181:
166: +
161:
142:
35:
28:
23:
22:
18:Orthic triangle
15:
12:
11:
5:
6721:
6711:
6710:
6696:
6695:
6685:
6679:
6673:
6652:
6651:External links
6649:
6648:
6647:
6642:
6629:
6624:
6611:
6589:
6584:
6571:
6557:
6554:
6552:
6551:
6476:
6413:
6386:(2): 249â260.
6363:
6335:
6322:
6303:
6300:
6296:
6292:
6287:
6284:
6280:
6276:
6271:
6268:
6264:
6247:
6234:
6218:
6205:
6193:
6181:
6165:
6159:
6141:
6128:
6116:
6104:
6092:
6080:
6068:
6060:
6035:
6019:
6003:
5991:
5979:
5967:
5955:
5940:
5928:
5917:
5905:
5893:
5874:
5862:
5834:
5822:
5805:
5793:Andrica, Dorin
5784:
5758:
5743:
5730:
5728:
5725:
5724:
5723:
5718:
5711:
5708:
5658:Book of Lemmas
5640:Book of Lemmas
5615:Book of Lemmas
5605:
5602:
5601:
5600:
5581:
5578:
5573:
5570:
5567:
5564:
5561:
5554:
5551:
5549:
5547:
5544:
5541:
5540:
5537:
5534:
5531:
5528:
5525:
5519:
5516:
5510:
5507:
5505:
5503:
5500:
5497:
5494:
5491:
5485:
5482:
5476:
5475:
5454:
5453:
5442:
5435:
5431:
5427:
5422:
5415:
5411:
5407:
5402:
5395:
5390:
5386:
5382:
5377:
5370:
5365:
5361:
5357:
5352:
5345:
5340:
5336:
5332:
5305:
5302:
5297:
5293:
5268:
5265:
5260:
5256:
5221:
5220:Right triangle
5218:
5201:
5198:
5196:
5193:
5192:
5191:
5175:
5171:
5167:
5162:
5155:
5151:
5147:
5142:
5135:
5131:
5127:
5102:
5099:
5096:
5093:
5090:
5068:
5064:
5060:
5054:
5051:
5045:
5040:
5036:
5032:
5026:
5023:
5017:
5012:
5008:
5004:
4998:
4995:
4980:
4977:
4976:
4975:
4964:
4959:
4953:
4949:
4946:
4939:
4934:
4928:
4924:
4921:
4914:
4909:
4903:
4899:
4896:
4889:
4884:
4878:
4874:
4871:
4835:
4832:
4831:
4830:
4819:
4814:
4809:
4806:
4801:
4797:
4793:
4790:
4787:
4784:
4779:
4776:
4771:
4767:
4763:
4760:
4757:
4754:
4749:
4746:
4741:
4737:
4733:
4730:
4727:
4724:
4719:
4716:
4711:
4708:
4703:
4700:
4697:
4694:
4666:
4660:
4657:
4652:
4648:
4644:
4639:
4636:
4631:
4627:
4623:
4618:
4615:
4610:
4606:
4598:
4595:
4582:
4578:
4574:
4564:
4561:
4560:
4559:
4548:
4545:
4538:
4534:
4528:
4524:
4518:
4511:
4507:
4501:
4497:
4491:
4484:
4480:
4474:
4470:
4451:
4444:
4437:
4424:
4417:
4410:
4402:
4401:Interior point
4399:
4398:
4397:
4386:
4380:
4377:
4372:
4369:
4363:
4358:
4354:
4321:
4315:
4312:
4311:
4310:
4298:
4291:
4287:
4283:
4278:
4271:
4267:
4263:
4258:
4251:
4247:
4243:
4238:
4233:
4230:
4203:
4199:
4195:
4185:
4182:
4176:
4169:
4154:By exchanging
4128:
4120:
4112:
4106:
4103:
4085:
4084:
4073:
4068:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4012:
4006:
4001:
3997:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3942:
3939:
3933:
3930:
3911:
3908:
3906:
3903:
3875:
3872:
3869:
3867:
3864:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3843:
3840:
3836:
3835:
3832:
3829:
3827:
3824:
3822:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3800:
3797:
3793:
3792:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3763:
3761:
3757:
3754:
3750:
3749:
3700:
3695:
3691:
3687:
3682:
3678:
3674:
3670:
3667:
3647:
3642:
3638:
3634:
3629:
3625:
3621:
3617:
3614:
3594:
3589:
3585:
3581:
3576:
3572:
3568:
3564:
3561:
3547:
3543:
3525:
3496:acute triangle
3485:extended sides
3466:
3463:
3461:
3458:
3456:
3453:
3450:
3447:
3445:
3442:
3440:
3437:
3434:
3431:
3429:
3426:
3423:
3422:
3419:
3416:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3387:
3384:
3382:
3379:
3376:
3375:
3372:
3369:
3366:
3363:
3361:
3358:
3356:
3353:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3329:
3328:
3282:pedal triangle
3238:
3235:
3234:
3233:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3186:
3182:
3178:
3175:
3170:
3166:
3162:
3159:
3156:
3154:
3150:
3144:
3140:
3137:
3130:
3129:
3126:
3123:
3118:
3114:
3110:
3105:
3101:
3097:
3092:
3088:
3084:
3081:
3076:
3072:
3068:
3065:
3062:
3060:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3026:
3022:
3018:
3015:
3010:
3006:
3002:
2999:
2997:
2993:
2987:
2983:
2980:
2973:
2972:
2932:
2931:
2916:
2911:
2907:
2904:
2898:
2895:
2893:
2889:
2885:
2882:
2876:
2875:
2872:
2867:
2863:
2860:
2854:
2851:
2849:
2845:
2841:
2838:
2832:
2831:
2810:
2809:
2794:
2789:
2785:
2782:
2776:
2771:
2767:
2764:
2758:
2755:
2753:
2750:
2745:
2741:
2738:
2732:
2729:
2724:
2720:
2717:
2711:
2709:
2657:Main article:
2654:
2651:
2628:
2627:
2616:
2611:
2607:
2604:
2598:
2593:
2589:
2586:
2543:
2542:
2527:
2522:
2518:
2514:
2511:
2508:
2505:
2500:
2494:
2490:
2487:
2480:
2475:
2469:
2465:
2462:
2455:
2450:
2444:
2440:
2437:
2430:
2425:
2421:
2417:
2412:
2407:
2403:
2399:
2394:
2389:
2385:
2381:
2376:
2371:
2367:
2363:
2361:
2358:
2355:
2352:
2349:
2344:
2340:
2337:
2331:
2326:
2322:
2319:
2313:
2308:
2304:
2301:
2295:
2292:
2289:
2284:
2280:
2276:
2271:
2267:
2263:
2258:
2254:
2250:
2248:
2224:
2220:
2216:
2203:
2202:
2191:
2186:
2182:
2178:
2175:
2170:
2164:
2160:
2157:
2150:
2145:
2139:
2135:
2132:
2125:
2120:
2114:
2110:
2107:
2100:
2095:
2091:
2087:
2082:
2078:
2074:
2069:
2065:
2041:
2038:
2037:
2036:
2029:
2014:
2002:
2001:
1990:
1987:
1981:
1977:
1974:
1968:
1964:
1961:
1954:
1948:
1944:
1941:
1935:
1931:
1928:
1921:
1915:
1911:
1908:
1902:
1898:
1895:
1880:
1879:
1875:
1874:
1863:
1860:
1854:
1850:
1847:
1841:
1837:
1834:
1827:
1821:
1817:
1814:
1808:
1804:
1801:
1794:
1788:
1784:
1781:
1775:
1771:
1768:
1753:
1752:
1740:
1739:
1728:
1717:
1712:
1708:
1705:
1699:
1694:
1690:
1687:
1681:
1676:
1672:
1669:
1663:
1658:
1654:
1651:
1645:
1640:
1636:
1633:
1627:
1622:
1618:
1615:
1601:
1600:
1584:
1581:
1575:, proposed by
1569:
1568:
1557:
1551:
1547:
1544:
1532:
1529:
1526:
1523:
1520:
1517:
1510:
1506:
1500:
1496:
1493:
1486:
1483:
1479:
1473:
1469:
1466:
1454:
1451:
1448:
1445:
1442:
1439:
1432:
1428:
1422:
1418:
1415:
1378:
1377:
1364:
1360:
1356:
1351:
1347:
1343:
1338:
1334:
1330:
1325:
1321:
1293:
1289:
1285:
1279:represent the
1262:right triangle
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1197:
1195:
1192:
1187:
1183:
1179:
1174:
1170:
1166:
1161:
1157:
1153:
1150:
1147:
1142:
1138:
1134:
1129:
1125:
1121:
1116:
1112:
1108:
1105:
1102:
1097:
1093:
1089:
1084:
1080:
1076:
1071:
1067:
1063:
1060:
1057:
1052:
1048:
1044:
1039:
1035:
1031:
1026:
1022:
1018:
1015:
1012:
1007:
1003:
999:
994:
990:
986:
981:
977:
973:
970:
965:
961:
957:
952:
948:
944:
939:
935:
931:
928:
926:
924:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
789:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
749:
723:
718:
714:
711:
705:
701:
698:
695:
691:
686:
682:
679:
673:
669:
666:
663:
659:
654:
650:
647:
641:
637:
634:
616:if and only if
589:
586:
566:
565:
539:
536:
531:
526:
522:
505:
485:right triangle
463:
460:
454:
450:
447:
441:
437:
433:
430:
427:
425:
419:
416:
413:
410:
409:
404:
398:
394:
389:
382:
378:
371:
366:
360:
356:
351:
344:
340:
333:
330:
328:
324:
320:
315:
311:
308:
305:
301:
294:
290:
286:
285:
280:
276:
271:
263:
259:
254:
251:
249:
243:
239:
235:
232:
229:
226:
223:
222:
213:
136:angle bisector
43:acute triangle
26:
9:
6:
4:
3:
2:
6720:
6709:
6706:
6705:
6703:
6693:
6689:
6686:
6683:
6680:
6677:
6674:
6669:
6668:
6663:
6660:
6655:
6654:
6645:
6643:0-534-35188-3
6639:
6635:
6630:
6627:
6621:
6617:
6612:
6602:
6598:
6594:
6590:
6587:
6585:0-13-087121-4
6581:
6577:
6572:
6568:
6564:
6560:
6559:
6546:
6541:
6537:
6533:
6529:
6525:
6517:
6513:
6505:
6501:
6494:
6490:
6486:
6480:
6466:
6462:
6458:
6453:
6448:
6444:
6440:
6436:
6435:
6427:
6423:
6417:
6409:
6405:
6401:
6393:
6389:
6385:
6381:
6374:
6367:
6358:
6353:
6348:
6347:Newton, Isaac
6342:
6340:
6332:
6326:
6319:
6301:
6298:
6294:
6290:
6285:
6282:
6278:
6274:
6269:
6266:
6262:
6251:
6244:
6238:
6231:
6227:
6222:
6215:
6209:
6202:
6197:
6190:
6185:
6179:
6175:
6169:
6162:
6160:0-06-500006-4
6156:
6152:
6145:
6138:
6132:
6125:
6120:
6114:, pp. 120-122
6113:
6108:
6101:
6096:
6089:
6084:
6077:
6072:
6063:
6057:
6053:
6049:
6042:
6040:
6032:
6026:
6024:
6017:
6013:
6007:
6001:, pp. 124-126
6000:
5995:
5988:
5983:
5977:
5971:
5965:
5959:
5953:
5951:
5944:
5937:
5932:
5926:
5921:
5915:
5909:
5902:
5897:
5890:
5886:
5881:
5879:
5871:
5866:
5852:on 2012-07-05
5851:
5847:
5841:
5839:
5831:
5826:
5819:
5818:0-486-61348-8
5815:
5809:
5802:
5798:
5794:
5788:
5774:on 2012-04-19
5773:
5769:
5762:
5755:
5750:
5748:
5740:
5735:
5731:
5722:
5719:
5717:
5714:
5713:
5707:
5705:
5701:
5697:
5693:
5688:
5686:
5675:
5671:
5667:
5663:
5659:
5654:
5645:
5641:
5637:
5629:
5625:
5621:
5617:
5616:
5611:
5579:
5576:
5571:
5568:
5565:
5562:
5559:
5552:
5550:
5545:
5542:
5535:
5532:
5529:
5526:
5523:
5517:
5514:
5508:
5506:
5501:
5498:
5495:
5492:
5489:
5483:
5480:
5466:
5465:
5464:
5461:
5459:
5440:
5433:
5429:
5425:
5420:
5413:
5409:
5405:
5400:
5393:
5388:
5384:
5380:
5375:
5368:
5363:
5359:
5355:
5350:
5343:
5338:
5334:
5330:
5321:
5320:
5319:
5303:
5300:
5295:
5291:
5266:
5263:
5258:
5254:
5226:
5217:
5215:
5211:
5195:Special cases
5173:
5169:
5165:
5160:
5153:
5149:
5145:
5140:
5133:
5129:
5125:
5116:
5115:
5114:
5100:
5097:
5094:
5091:
5088:
5066:
5062:
5058:
5052:
5049:
5043:
5038:
5034:
5030:
5024:
5021:
5015:
5010:
5006:
5002:
4996:
4993:
4962:
4957:
4947:
4944:
4937:
4932:
4922:
4919:
4912:
4907:
4897:
4894:
4887:
4882:
4872:
4869:
4858:
4857:
4856:
4853:
4817:
4807:
4804:
4799:
4795:
4791:
4788:
4777:
4774:
4769:
4765:
4761:
4758:
4747:
4744:
4739:
4735:
4731:
4728:
4722:
4717:
4714:
4709:
4706:
4683:
4682:
4681:
4664:
4658:
4655:
4650:
4646:
4642:
4637:
4634:
4629:
4625:
4621:
4616:
4613:
4608:
4604:
4596:
4593:
4546:
4543:
4536:
4532:
4526:
4522:
4516:
4509:
4505:
4499:
4495:
4489:
4482:
4478:
4472:
4468:
4458:
4457:
4456:
4450:
4443:
4436:
4423:
4416:
4409:
4384:
4378:
4375:
4370:
4367:
4361:
4356:
4352:
4344:
4343:
4342:
4336:
4296:
4289:
4285:
4281:
4276:
4269:
4265:
4261:
4256:
4249:
4245:
4241:
4236:
4231:
4228:
4218:
4217:
4216:
4210:
4181:
4152:
4126:
4118:
4110:
4104:
4101:
4090:
4071:
4066:
4057:
4054:
4051:
4042:
4039:
4036:
4027:
4024:
4021:
4015:
4010:
4004:
3999:
3995:
3987:
3986:
3985:
3967:
3961:
3958:
3955:
3952:
3949:
3940:
3937:
3931:
3928:
3921:
3920:semiperimeter
3902:
3900:
3895:
3893:
3873:
3870:
3865:
3860:
3855:
3850:
3845:
3841:
3838:
3830:
3825:
3820:
3817:
3812:
3807:
3802:
3798:
3795:
3787:
3782:
3777:
3772:
3767:
3764:
3759:
3755:
3752:
3738:
3736:
3732:
3728:
3723:
3716:
3698:
3693:
3689:
3685:
3680:
3676:
3672:
3668:
3665:
3645:
3640:
3636:
3632:
3627:
3623:
3619:
3615:
3612:
3592:
3587:
3583:
3579:
3574:
3570:
3566:
3562:
3559:
3540:, and define
3534:
3521:
3516:
3513:
3506:
3503:
3501:
3497:
3492:
3490:
3486:
3481:
3464:
3459:
3454:
3451:
3448:
3443:
3438:
3435:
3432:
3427:
3424:
3417:
3414:
3411:
3406:
3401:
3396:
3391:
3388:
3385:
3380:
3377:
3370:
3367:
3364:
3359:
3354:
3351:
3348:
3343:
3338:
3333:
3330:
3318:
3314:
3311:
3304:
3297:
3291:
3287:
3283:
3279:
3274:
3264:
3257:
3250:
3243:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3184:
3180:
3176:
3173:
3168:
3164:
3160:
3157:
3155:
3148:
3138:
3135:
3124:
3116:
3112:
3108:
3103:
3099:
3095:
3090:
3086:
3079:
3074:
3070:
3066:
3063:
3061:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3024:
3020:
3016:
3013:
3008:
3004:
3000:
2998:
2991:
2981:
2978:
2963:
2962:
2961:
2956:
2949:
2914:
2905:
2902:
2896:
2894:
2883:
2880:
2870:
2861:
2858:
2852:
2850:
2839:
2836:
2822:
2821:
2820:
2815:
2792:
2783:
2780:
2774:
2765:
2762:
2756:
2748:
2739:
2736:
2730:
2727:
2718:
2715:
2700:
2699:
2698:
2696:
2692:
2688:
2677:
2670:
2660:
2650:
2648:
2644:
2639:
2637:
2633:
2614:
2605:
2602:
2596:
2587:
2584:
2574:
2573:
2572:
2525:
2520:
2512:
2509:
2503:
2498:
2488:
2485:
2478:
2473:
2463:
2460:
2453:
2448:
2438:
2435:
2428:
2423:
2419:
2415:
2410:
2405:
2401:
2397:
2392:
2387:
2383:
2379:
2374:
2369:
2365:
2356:
2353:
2350:
2347:
2338:
2335:
2329:
2320:
2317:
2311:
2302:
2299:
2293:
2290:
2287:
2282:
2278:
2274:
2269:
2265:
2261:
2256:
2252:
2239:
2238:
2237:
2231:
2212:
2189:
2184:
2180:
2176:
2173:
2168:
2158:
2155:
2148:
2143:
2133:
2130:
2123:
2118:
2108:
2105:
2098:
2093:
2089:
2085:
2080:
2076:
2072:
2067:
2063:
2055:
2054:
2053:
2047:
2034:
2030:
2027:
2023:
2019:
2015:
2012:
2008:
2004:
2003:
1988:
1985:
1975:
1972:
1962:
1959:
1952:
1942:
1939:
1929:
1926:
1919:
1909:
1906:
1896:
1893:
1882:
1881:
1877:
1876:
1861:
1858:
1848:
1845:
1835:
1832:
1825:
1815:
1812:
1802:
1799:
1792:
1782:
1779:
1769:
1766:
1755:
1754:
1750:
1746:
1742:
1741:
1737:
1729:
1715:
1706:
1703:
1697:
1688:
1685:
1679:
1670:
1667:
1661:
1652:
1649:
1643:
1634:
1631:
1625:
1616:
1613:
1603:
1602:
1598:
1597:
1596:
1580:
1578:
1574:
1555:
1545:
1542:
1504:
1494:
1491:
1484:
1481:
1477:
1467:
1464:
1426:
1416:
1413:
1402:
1401:
1400:
1398:
1389:
1362:
1358:
1354:
1349:
1345:
1341:
1336:
1332:
1328:
1323:
1319:
1311:
1310:
1309:
1306:
1300:
1282:
1274:
1273:complex plane
1269:
1267:
1263:
1259:
1253:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1198:
1185:
1181:
1177:
1172:
1168:
1164:
1159:
1155:
1151:
1140:
1136:
1132:
1127:
1123:
1119:
1114:
1110:
1103:
1095:
1091:
1087:
1082:
1078:
1074:
1069:
1065:
1061:
1050:
1046:
1042:
1037:
1033:
1029:
1024:
1020:
1013:
1005:
1001:
997:
992:
988:
984:
979:
975:
963:
959:
955:
950:
946:
942:
937:
933:
927:
914:
913:
908:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
790:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
739:
738:
721:
712:
709:
703:
699:
696:
693:
689:
680:
677:
671:
667:
664:
661:
657:
648:
645:
639:
635:
632:
619:
617:
609:
600:
595:
585:
583:
582:extended side
579:
570:
563:
559:
558:Special Cases
555:
537:
534:
529:
524:
520:
512:
511:
510:
490:
486:
461:
458:
452:
448:
445:
439:
435:
431:
428:
426:
417:
414:
411:
402:
396:
392:
387:
380:
376:
369:
364:
358:
354:
349:
342:
338:
331:
329:
322:
318:
313:
309:
306:
303:
299:
292:
288:
278:
274:
269:
261:
257:
252:
250:
241:
233:
230:
227:
209:
205:
201:
193:
189:
185:
177:
173:
169:
165:
159:
154:
150:
148:
139:
137:
133:
129:
125:
120:
118:
114:
109:
107:
103:
98:
94:
93:
92:extended base
88:
87:
82:
78:
77:perpendicular
74:
73:
68:
64:
60:
56:
52:
44:
39:
33:
19:
6665:
6633:
6615:
6604:. Retrieved
6601:Cut the Knot
6600:
6575:
6566:
6535:
6531:
6515:
6503:
6492:
6488:
6479:
6468:. Retrieved
6465:Cut The Knot
6464:
6454:. Quoted by
6438:
6432:
6416:
6407:
6403:
6383:
6379:
6366:
6356:
6330:
6325:
6317:
6250:
6242:
6237:
6229:
6221:
6213:
6208:
6201:Johnson 2007
6196:
6189:Johnson 2007
6184:
6173:
6168:
6150:
6144:
6136:
6131:
6124:Johnson 2007
6119:
6107:
6100:Johnson 2007
6095:
6083:
6076:Johnson 2007
6071:
6051:
6030:
6011:
6006:
5994:
5982:
5970:
5958:
5949:
5943:
5931:
5920:
5908:
5896:
5870:Johnson 2007
5865:
5854:. Retrieved
5850:the original
5830:Johnson 2007
5825:
5808:
5787:
5776:. Retrieved
5772:the original
5761:
5734:
5689:
5673:
5670:Isaac Newton
5668:(1619), and
5665:
5657:
5655:
5639:
5627:
5613:
5607:
5462:
5455:
5231:
5203:
4982:
4851:
4837:
4566:
4563:Area theorem
4448:
4441:
4434:
4421:
4414:
4407:
4404:
4335:circumradius
4317:
4187:
4153:
4086:
3913:
3896:
3739:
3721:
3714:
3532:
3517:
3511:
3507:
3504:
3493:
3482:
3315:
3309:
3302:
3295:
3289:
3285:
3272:
3268:
3262:
3255:
3248:
2955:circumradius
2933:
2811:
2676:circumcenter
2662:
2640:
2629:
2544:
2204:
2046:circumradius
2043:
2011:circumcenter
1751:through it.)
1736:polar circle
1586:
1572:
1570:
1397:free vectors
1395:by means of
1387:
1379:
1304:
1301:of triangle
1299:circumcenter
1270:
1254:
915:
909:
740:
620:
607:
605:
578:obtuse angle
575:
482:
207:
203:
199:
191:
187:
183:
175:
171:
167:
163:
146:
140:
121:
110:
101:
96:
90:
84:
70:
63:line segment
54:
48:
5630:, VII, 62;
2643:circumconic
2632:directrices
2044:Denote the
608:orthocenter
588:Orthocenter
6662:"Altitude"
6606:2022-12-17
6556:References
6470:2019-11-17
5901:Smart 1998
5856:2012-05-04
5820:, page 142
5778:2012-04-19
5739:Smart 1998
5620:Archimedes
5208:within an
3735:Euler line
3727:homothetic
3536:at vertex
2691:Euler line
2555:, so that
1583:Properties
592:See also:
489:hypotenuse
6667:MathWorld
6618:, Dover,
6565:(2007) ,
6538:: 60â96.
6512:"ZusÀtze"
6299:−
6283:−
6267:−
5666:Geometrie
5636:al-Nasawi
5566:⋅
5530:⋅
5496:⋅
5113:implies
4952:¯
4927:¯
4902:¯
4877:¯
4805:−
4792:−
4775:−
4762:−
4745:−
4732:−
4707:−
4656:−
4635:−
4614:−
4119:×
4111:×
4055:−
4040:−
4025:−
3871:−
3818:−
3765:−
3686:∩
3633:∩
3580:∩
3452:
3436:
3415:
3389:
3368:
3352:
3245:Triangle
3211:
3202:
3193:
3174:−
3143:¯
3080:−
3051:
3042:
3033:
3014:−
2986:¯
2910:¯
2888:¯
2866:¯
2844:¯
2788:¯
2770:¯
2744:¯
2723:¯
2636:parabolas
2610:¯
2592:¯
2493:¯
2468:¯
2443:¯
2343:¯
2325:¯
2307:¯
2230:excircles
2163:¯
2138:¯
2113:¯
1980:¯
1967:¯
1947:¯
1934:¯
1914:¯
1901:¯
1853:¯
1840:¯
1820:¯
1807:¯
1787:¯
1774:¯
1711:¯
1698:⋅
1693:¯
1675:¯
1662:⋅
1657:¯
1639:¯
1626:⋅
1621:¯
1550:→
1509:∑
1499:→
1485:⋅
1472:→
1431:∑
1421:→
1231:
1219:
1207:
1152:−
1120:−
1062:−
1043:−
985:−
956:−
886:
877:
871:−
865:
853:
844:
838:−
832:
820:
811:
805:−
799:
780:
768:
756:
717:¯
685:¯
653:¯
446:∴
403:⏞
365:⏞
128:congruent
6702:Category
6526:(1883).
6487:(1804).
6459:(2010).
6424:(1850).
6349:(1971).
6090:, p. 165
5989:, p. 123
5938:, p. 102
5903:, p. 182
5756:, p. 118
5741:, p. 156
5710:See also
4680:we have
4209:incircle
3842:″
3799:″
3756:″
3669:″
3616:″
3563:″
2948:inradius
2814:incenter
2695:midpoint
2669:centroid
2211:incircle
552: (
210: )
194: )
178: )
132:midpoint
69:(called
59:triangle
55:altitude
51:geometry
6569:, Dover
6410:: 1â36.
5644:al-Quhi
5604:History
5316:
5283:
5279:
5246:
4855:, then
4569:a, b, c
4211:radius
4190:a, b, c
3916:a, b, c
3494:In any
3278:oblique
2685:of the
2634:of all
2052:. Then
2024:of the
1749:cevians
1593:A, B, C
1589:D, E, F
1281:numbers
1277:A, B, C
1271:In the
623:A, B, C
145:(as in
6640:
6622:
6582:
6157:
6058:
5816:
5799:
5683:Later
5681:1680).
5624:Pappus
4123:height
3715:A"B"C"
2674:, the
2667:, the
2232:, and
556:; see
147:height
122:In an
75:) and
67:vertex
6516:Werke
6491:[
6437:. 3.
6429:(PDF)
6376:(PDF)
5727:Notes
4158:with
483:In a
61:is a
57:of a
53:, an
6638:ISBN
6620:ISBN
6580:ISBN
6521:See
6155:ISBN
6056:ISBN
5814:ISBN
5797:ISBN
5281:and
5236:and
5141:<
5092:<
4329:and
4173:and
4115:base
3918:and
3899:here
3483:The
2953:and
2897:>
2853:<
2630:The
2016:The
2005:The
1587:Let
910:and
621:Let
498:and
196:and
97:foot
86:base
81:edge
72:apex
6540:doi
6443:doi
6388:doi
6316:,"
5653:).
5648:fl.
4852:ABC
4838:If
4581:, h
4577:, h
4405:If
4202:, h
4198:, h
4162:or
4151:).
3722:ABC
3546:, L
3533:ABC
3512:ABC
3449:sec
3433:sec
3412:sec
3386:sec
3365:sec
3349:sec
3310:ABC
3303:DEF
3296:DEF
3288:or
3276:is
3273:ABC
3263:ABC
3256:DEF
3249:abc
3208:cos
3199:cos
3190:cos
3048:cos
3039:cos
3030:cos
2223:, r
2219:, r
1388:ABC
1305:ABC
1292:, z
1288:, z
1228:tan
1216:tan
1204:tan
883:sin
874:sin
862:cos
850:sin
841:sin
829:cos
817:sin
808:sin
796:cos
777:sec
765:sec
753:sec
49:In
6704::
6664:.
6599:.
6595:.
6534:.
6530:.
6514:.
6463:.
6439:37
6431:.
6406:.
6402:.
6384:60
6382:.
6378:.
6338:^
6050:.
6038:^
6022:^
5887:,
5877:^
5837:^
5746:^
5679:c.
5632:c.
5460:.
5216:.
4845:AD
4547:1.
4447:,
4440:,
4420:,
4413:,
3901:.
3894:.
3737:.
3491:.
3313:.
2960:,
2946:,
2942:,
2938:,
2649:.
2641:A
2571::
2568:HP
2558:AD
2548:AD
2213:,
2177:12
1989:2.
1862:1.
1579:.
1268:.
560:,
206:,
202:,
190:,
186:,
180:,
174:,
170:,
119:.
108:.
6694:.
6670:.
6609:.
6548:.
6542::
6536:1
6473:.
6449:.
6445::
6408:6
6394:.
6390::
6302:2
6295:d
6291:=
6286:2
6279:b
6275:+
6270:2
6263:a
6064:.
5891:.
5859:.
5781:.
5677:(
5646:(
5626:(
5580:B
5577:A
5572:C
5569:B
5563:C
5560:A
5553:=
5546:D
5543:C
5536:D
5533:C
5527:B
5524:A
5518:2
5515:1
5509:=
5502:C
5499:B
5493:C
5490:A
5484:2
5481:1
5441:.
5434:2
5430:b
5426:1
5421:+
5414:2
5410:a
5406:1
5401:=
5394:2
5389:b
5385:h
5381:1
5376:+
5369:2
5364:a
5360:h
5356:1
5351:=
5344:2
5339:c
5335:h
5331:1
5304:a
5301:=
5296:b
5292:h
5267:b
5264:=
5259:a
5255:h
5242:c
5238:b
5234:a
5206:P
5190:.
5174:c
5170:h
5166:1
5161:+
5154:b
5150:h
5146:1
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5130:h
5126:1
5101:c
5098:+
5095:b
5089:a
5067:c
5063:h
5059:c
5053:2
5050:1
5044:=
5039:b
5035:h
5031:b
5025:2
5022:1
5016:=
5011:a
5007:h
5003:a
4997:2
4994:1
4963:.
4958:2
4948:E
4945:C
4938:+
4933:2
4923:B
4920:A
4913:=
4908:2
4898:B
4895:E
4888:+
4883:2
4873:C
4870:A
4850:âł
4840:E
4818:.
4813:)
4808:1
4800:c
4796:h
4789:H
4786:(
4783:)
4778:1
4770:b
4766:h
4759:H
4756:(
4753:)
4748:1
4740:a
4736:h
4729:H
4726:(
4723:H
4718:4
4715:=
4710:1
4702:a
4699:e
4696:r
4693:A
4665:2
4659:1
4651:c
4647:h
4643:+
4638:1
4630:b
4626:h
4622:+
4617:1
4609:a
4605:h
4597:=
4594:H
4583:c
4579:b
4575:a
4573:h
4544:=
4537:3
4533:h
4527:3
4523:p
4517:+
4510:2
4506:h
4500:2
4496:p
4490:+
4483:1
4479:h
4473:1
4469:p
4452:3
4449:h
4445:2
4442:h
4438:1
4435:h
4430:P
4425:3
4422:p
4418:2
4415:p
4411:1
4408:p
4385:.
4379:R
4376:2
4371:c
4368:b
4362:=
4357:a
4353:h
4339:R
4331:c
4327:b
4322:a
4320:h
4297:.
4290:c
4286:h
4282:1
4277:+
4270:b
4266:h
4262:1
4257:+
4250:a
4246:h
4242:1
4237:=
4232:r
4229:1
4213:r
4204:c
4200:b
4196:a
4194:h
4177:c
4175:h
4170:b
4168:h
4164:c
4160:b
4156:a
4149:a
4145:A
4141:a
4127:,
4105:2
4102:1
4072:.
4067:a
4061:)
4058:c
4052:s
4049:(
4046:)
4043:b
4037:s
4034:(
4031:)
4028:a
4022:s
4019:(
4016:s
4011:2
4005:=
4000:a
3996:h
3982:a
3968:,
3965:)
3962:c
3959:+
3956:b
3953:+
3950:a
3947:(
3941:2
3938:1
3932:=
3929:s
3874:c
3866::
3861:b
3856::
3851:a
3846:=
3839:C
3831:c
3826::
3821:b
3813::
3808:a
3803:=
3796:B
3788:c
3783::
3778:b
3773::
3768:a
3760:=
3753:A
3720:âł
3713:âł
3699:.
3694:A
3690:L
3681:C
3677:L
3673:=
3666:C
3646:,
3641:A
3637:L
3628:C
3624:L
3620:=
3613:B
3593:,
3588:C
3584:L
3575:B
3571:L
3567:=
3560:A
3548:C
3544:B
3542:L
3538:A
3531:âł
3526:A
3524:L
3510:âł
3465:0
3460::
3455:B
3444::
3439:A
3428:=
3425:F
3418:C
3407::
3402:0
3397::
3392:A
3381:=
3378:E
3371:C
3360::
3355:B
3344::
3339:0
3334:=
3331:D
3308:âł
3301:âł
3294:âł
3271:âł
3261:âł
3254:âł
3247:âł
3217:.
3214:C
3205:B
3196:A
3185:2
3181:R
3177:4
3169:2
3165:r
3161:2
3158:=
3149:2
3139:I
3136:H
3125:,
3122:)
3117:2
3113:c
3109:+
3104:2
3100:b
3096:+
3091:2
3087:a
3083:(
3075:2
3071:R
3067:9
3064:=
3054:C
3045:B
3036:A
3025:2
3021:R
3017:8
3009:2
3005:R
3001:=
2992:2
2982:H
2979:O
2958:R
2951:r
2944:c
2940:b
2936:a
2915:.
2906:G
2903:I
2884:G
2881:H
2871:,
2862:G
2859:H
2840:I
2837:H
2817:I
2793:.
2784:H
2781:G
2775:=
2766:G
2763:O
2757:2
2749:,
2740:H
2737:N
2731:2
2728:=
2719:H
2716:O
2683:N
2679:O
2672:G
2665:H
2615:.
2606:P
2603:D
2597:=
2588:D
2585:H
2563:D
2553:P
2526:.
2521:2
2517:)
2513:R
2510:2
2507:(
2504:+
2499:2
2489:H
2486:C
2479:+
2474:2
2464:H
2461:B
2454:+
2449:2
2439:H
2436:A
2429:=
2424:2
2420:r
2416:+
2411:2
2406:c
2402:r
2398:+
2393:2
2388:b
2384:r
2380:+
2375:2
2370:a
2366:r
2357:,
2354:R
2351:2
2348:+
2339:H
2336:C
2330:+
2321:H
2318:B
2312:+
2303:H
2300:A
2294:=
2291:r
2288:+
2283:c
2279:r
2275:+
2270:b
2266:r
2262:+
2257:a
2253:r
2234:R
2225:c
2221:b
2217:a
2215:r
2207:r
2190:.
2185:2
2181:R
2174:=
2169:2
2159:H
2156:C
2149:+
2144:2
2134:H
2131:B
2124:+
2119:2
2109:H
2106:A
2099:+
2094:2
2090:c
2086:+
2081:2
2077:b
2073:+
2068:2
2064:a
2050:R
2028:.
1986:=
1976:F
1973:C
1963:H
1960:C
1953:+
1943:E
1940:B
1930:H
1927:B
1920:+
1910:D
1907:A
1897:H
1894:A
1859:=
1849:F
1846:C
1836:F
1833:H
1826:+
1816:E
1813:B
1803:E
1800:H
1793:+
1783:D
1780:A
1770:D
1767:H
1738:.
1732:H
1716:.
1707:F
1704:H
1689:H
1686:C
1680:=
1671:E
1668:H
1653:H
1650:B
1644:=
1635:D
1632:H
1617:H
1614:A
1556:.
1546:A
1543:H
1531:c
1528:i
1525:l
1522:c
1519:y
1516:c
1505:=
1495:O
1492:H
1482:2
1478:,
1468:A
1465:O
1453:c
1450:i
1447:l
1444:c
1441:y
1438:c
1427:=
1417:H
1414:O
1393:H
1386:âł
1382:H
1363:C
1359:z
1355:+
1350:B
1346:z
1342:+
1337:A
1333:z
1329:=
1324:H
1320:z
1303:âł
1294:C
1290:B
1286:A
1284:z
1237:.
1234:C
1225::
1222:B
1213::
1210:A
1201:=
1191:)
1186:2
1182:c
1178:+
1173:2
1169:b
1165:+
1160:2
1156:a
1149:(
1146:)
1141:2
1137:c
1133:+
1128:2
1124:b
1115:2
1111:a
1107:(
1104::
1101:)
1096:2
1092:c
1088:+
1083:2
1079:b
1075:+
1070:2
1066:a
1059:(
1056:)
1051:2
1047:c
1038:2
1034:b
1030:+
1025:2
1021:a
1017:(
1014::
1011:)
1006:2
1002:c
998:+
993:2
989:b
980:2
976:a
972:(
969:)
964:2
960:c
951:2
947:b
943:+
938:2
934:a
930:(
892:,
889:B
880:A
868:C
859::
856:A
847:C
835:B
826::
823:C
814:B
802:A
793:=
783:C
774::
771:B
762::
759:A
722:|
713:B
710:A
704:|
700:=
697:c
694:,
690:|
681:A
678:C
672:|
668:=
665:b
662:,
658:|
649:C
646:B
640:|
636:=
633:a
612:H
564:)
538:q
535:p
530:=
525:c
521:h
506:c
504:h
500:q
496:p
492:c
462:q
459:p
453:=
449:h
440:2
436:h
432:2
429:=
418:q
415:p
412:2
397:2
393:q
388:+
381:2
377:h
370:+
359:2
355:h
350:+
343:2
339:p
332:=
323:2
319:q
314:+
310:q
307:p
304:2
300:+
293:2
289:p
279:2
275:s
270:+
262:2
258:r
253:=
242:2
238:)
234:q
231:+
228:p
225:(
212:,
208:q
204:h
200:s
198:(
192:h
188:p
184:r
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176:s
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