3231:
5214:
588:
142:
27:
467:
2529:
558:
1240:
3220:
3877:
3468:
205:
2231:
1555:
1244:
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
1988:
1861:
907:
5587:
895:
1715:
2918:
2796:
2955:
5440:
3311:
722:
104:: 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
462:{\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}}}
2189:
4297:
3732:
4962:
573:, 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.
2524:{\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}}}
4817:
1394:
4546:
6361:
732:
5068:
5177:
4126:
1874:
1747:
5458:
4667:
4071:
1595:
88:
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
5463:
2814:
2692:
1235:{\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}}}
3215:{\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}}}
2960:
2819:
2697:
2236:
912:
737:
210:
5313:
2614:
5687:(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
3698:
3645:
3592:
3967:
1732:
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
4384:
2686:
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:
1364:
617:
561:
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.
3463:{\displaystyle {\begin{array}{rccccc}D=&0&:&\sec B&:&\sec C\\E=&\sec A&:&0&:&\sec C\\F=&\sec A&:&\sec B&:&0\end{array}}}
569:), 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
6303:
539:
2047:
4210:
4850:
5100:
5303:
5266:
1550:{\displaystyle {\vec {OH}}=\sum \limits _{\scriptstyle {\rm {cyclic}}}{\vec {OA}},\qquad 2\cdot {\vec {HO}}=\sum \limits _{\scriptstyle {\rm {cyclic}}}{\vec {HA}}.}
6696:
4675:
145:
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
3872:{\displaystyle {\begin{array}{rrcrcr}A''=&-a&:&b&:&c\\B''=&a&:&-b&:&c\\C''=&a&:&b&:&-c\end{array}}}
4450:
4975:
5108:
5611:(3rd century BC), citing the "commentary to the treatise about right-angled triangles", a work which does not survive. It was also mentioned by
4083:
6612:
6048:
5789:
1983:{\displaystyle {\frac {\overline {AH}}{\overline {AD}}}+{\frac {\overline {BH}}{\overline {BE}}}+{\frac {\overline {CH}}{\overline {CF}}}=2.}
1856:{\displaystyle {\frac {\overline {HD}}{\overline {AD}}}+{\frac {\overline {HE}}{\overline {BE}}}+{\frac {\overline {HF}}{\overline {CF}}}=1.}
900:
5582:{\displaystyle {\begin{aligned}{\tfrac {1}{2}}AC\cdot BC&={\tfrac {1}{2}}AB\cdot CD\\CD&={\tfrac {AC\cdot BC}{AB}}\\\end{aligned}}}
890:{\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}}}
5636:
5801:
Dörrie, Heinrich, "100 Great
Problems of Elementary Mathematics. Their History and Solution". Dover Publications, Inc., New York, 1965,
3491:, posed in 1775. The sides of the orthic triangle are parallel to the tangents to the circumcircle at the original triangle's vertices.
4578:
5756:
3979:
2627:
that are externally tangent to one side of a triangle and tangent to the extensions of the other sides pass through the orthocenter.
1710:{\displaystyle {\overline {AH}}\cdot {\overline {HD}}={\overline {BH}}\cdot {\overline {HE}}={\overline {CH}}\cdot {\overline {HF}}.}
6630:
6572:
6147:
5806:
2913:{\displaystyle {\begin{aligned}{\overline {HI}}&<{\overline {HG}},\\{\overline {HG}}&>{\overline {IG}}.\end{aligned}}}
2225:
again as the radius of its circumcircle, the following relations hold regarding the distances of the orthocenter from the vertices:
1867:
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:
565:
For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an
2791:{\displaystyle {\begin{aligned}&{\overline {OH}}=2{\overline {NH}},\\&2{\overline {OG}}={\overline {GH}}.\end{aligned}}}
6680:
1588:
The product of the lengths of the segments that the orthocenter divides an altitude into is the same for all three altitudes:
5435:{\displaystyle {\frac {1}{h_{c}^{2}}}={\frac {1}{h_{a}^{2}}}+{\frac {1}{h_{b}^{2}}}={\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}.}
2020:
Four points in the plane, such that one of them is the orthocenter of the triangle formed by the other three, is called an
5624:
5649:, but was not widely known in Europe, and the theorem was therefore proven several more times in the 17thâ19th century.
2566:
5446:
550:
607:
the triangle is acute. If one angle is a right angle, the orthocenter coincides with the vertex at the right angle.
5680:
5673:
5597:
The theorem that the three altitudes of a triangle concur (at the orthocenter) is not directly stated in surviving
3266:
31:
5834:
3715:
2620:
5873:
3913:
717:{\displaystyle a=\left|{\overline {BC}}\right|,b=\left|{\overline {CA}}\right|,c=\left|{\overline {AB}}\right|}
4336:
3316:
1303:
6551:
6473:
1724:
105:
5999:
Marie-Nicole Gras, "Distances between the circumcenter of the extouch triangle and the classical centers",
6422:
6410:
6246:
5692:
3737:
3649:
3596:
3543:
1565:
542:
146:
2184:{\displaystyle a^{2}+b^{2}+c^{2}+{\overline {AH}}^{2}+{\overline {BH}}^{2}+{\overline {CH}}^{2}=12R^{2}.}
6214:
5877:
3305:
2635:
725:
504:
116:
94:
5201:, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. This is
4292:{\displaystyle \displaystyle {\frac {1}{r}}={\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}.}
3230:
6512:
6488:
5684:
5612:
5202:
5198:
3880:
3719:
3504:
are parallel to the sides of the orthic triangle, forming a triangle similar to the orthic triangle.
3488:
3487:, the inscribed triangle with the smallest perimeter is the orthic triangle. This is the solution to
1287:
4957:{\displaystyle {\overline {AC}}^{2}+{\overline {EB}}^{2}={\overline {AB}}^{2}+{\overline {CE}}^{2}.}
6581:
6445:
5781:
3887:
3508:
2808:
than it is to the centroid, and the orthocenter is farther than the incenter is from the centroid:
2021:
582:
20:
6440:
6340:
2006:
1995:
112:
4077:
123:
of that side as its foot. Also the altitude having the incongruent side as its base will be the
6161:
Dorin
Andrica and Dan S ̧tefan Marinescu. "New Interpolation Inequalities to Euler's R ℠2r".
6647:
6626:
6608:
6568:
6500:
6143:
6044:
6036:
5802:
5785:
5709:
5645:
This proof in Arabic was translated as part of the (early 17th century) Latin editions of the
5598:
5213:
5073:
2675:
2647:
55:
6004:
5275:
5238:
6528:
6431:
6376:
6339:
5632:
4812:{\displaystyle \mathrm {Area} ^{-1}=4{\sqrt {H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}}.}
3477:
1385:
3476:
of the orthic triangle meet the opposite extended sides of its reference triangle at three
5760:
5704:
5688:
5650:
2014:
1254:
74:
69:
60:
6389:"Two beautiful geometrical theorems by Abƫ Sahl Kƫhī in a 17th century Dutch translation"
6585:
6166:
5603:
4541:{\displaystyle {\frac {p_{1}}{h_{1}}}+{\frac {p_{2}}{h_{2}}}+{\frac {p_{3}}{h_{3}}}=1.}
3484:
3270:
1269:
1250:
1246:
604:
473:
124:
6664:
6449:
6414:
5936:
5902:
3281:. That is, the feet of the altitudes of an oblique triangle form the orthic triangle,
84:
of the altitude. The intersection of the extended base and the altitude is called the
6690:
6676:
5964:
5695:
of the new triangle, and therefore concur (at the circumcenter of the new triangle).
3908:
3473:
2010:
1261:
570:
80:
65:
6230:
Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle",
6201:
Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle,"
5952:
587:
141:
26:
6670:
6335:
5658:
4323:
2943:
2664:
2034:
1999:
566:
51:
5937:
Bell, Amy, "Hansen's right triangle theorem, its converse and a generalization",
5063:{\displaystyle {\tfrac {1}{2}}ah_{a}={\tfrac {1}{2}}bh_{b}={\tfrac {1}{2}}ch_{c}}
3288:. Also, the incenter (the center of the inscribed circle) of the orthic triangle
5963:
Weisstein, Eric W. "Jerabek
Hyperbola." From MathWorld--A Wolfram Web Resource.
5913:
5901:
Weisstein, Eric W. "Isotomic conjugate" From MathWorld--A Wolfram Web
Resource.
2631:
595:
The three (possibly extended) altitudes intersect in a single point, called the
19:"Orthocenter" and "Orthocentre" redirect here. For the orthocentric system, see
5951:
Weisstein, Eric W. "Kiepert
Parabola." From MathWorld--A Wolfram Web Resource.
3729:
Trilinear coordinates for the vertices of the tangential triangle are given by
6650:
6533:
6516:
6435:
6380:
5608:
3723:
2679:
477:
3718:
to the orthic triangle. The circumcenter of the tangential triangle, and the
30:
The three altitudes of a triangle intersect at the orthocenter, which for an
6655:
3497:
The tangent lines of the nine-point circle at the midpoints of the sides of
5172:{\displaystyle {\frac {1}{h_{a}}}<{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}}
119:
sides), the altitude having the incongruent side as its base will have the
5217:
Comparison of the inverse
Pythagorean theorem with the Pythagorean theorem
4121:{\displaystyle {\tfrac {1}{2}}\times {\text{base}}\times {\text{height}},}
5914:
Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Web
Resource.
4197:
3494:
The orthic triangle of an acute triangle gives a triangular light route.
2936:
2802:
2683:
2657:
2624:
2218:
2199:
138:), often subscripted with the name of the side the altitude is drawn to.
120:
101:
47:
39:
5691:
of the new triangle, and the altitudes of the original triangle are the
5838:
4080:
for the area of a triangle in terms of the sides with the area formula
5874:
Panapoi, Ronnachai, "Some properties of the orthocenter of a triangle"
557:
6388:
1737:
1733:
16:
Perpendicular line segment from a triangle's side to opposite vertex
4575:, and denoting the semi-sum of the reciprocals of the altitudes as
78:). This (infinite) line containing the (finite) base is called the
6478:
Solutions peu connues de différens problÚmes de Géométrie-pratique
6018:
Smith, Geoff, and
Leversha, Gerry, "Euler and triangle geometry",
5212:
586:
556:
140:
25:
1723:
having radius the square root of this constant is the triangle's
1560:
The first of the previous vector identities is also known as the
614:
denote the vertices and also the angles of the triangle, and let
4662:{\displaystyle H={\tfrac {h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1}}{2}}}
1380:. From this, the following characterizations of the orthocenter
1297:
is located at the origin of the plane. Then, the complex number
4066:{\displaystyle h_{a}={\frac {2{\sqrt {s(s-a)(s-b)(s-c)}}}{a}}.}
5784:, "Complex numbers from A to...Z". BirkhÀuser, Boston, 2006,
6482:
Little-known solutions of various
Geometry practice problems
6348:. Vol. 4. Cambridge University Press. pp. 454â455.
6318:
Richinick, Jennifer, "The upside-down
Pythagorean Theorem,"
6517:"The Triangle and its Six Scribed Circles §5. Orthocentre"
6507:. Vol. 4. Göttingen Academy of Sciences. p. 396.
6495:. By Carnot, Lazare (in German). Translated by Schumacher.
6393:
TÄrÄ«kÍh-e ÊŸElm: Iranian Journal for the History of Science
6005:
http://forumgeom.fau.edu/FG2014volume14/FG201405index.html
3273:
of the orthocenter of the original triangle is called the
6484:] (in French). Devilly, Metz et Courcier. p. 15.
6450:"A Possibly First Proof of the Concurrence of Altitudes"
6124:
Bryant, V., and Bradley, H., "Triangular Light Routes,"
5623:
340). The theorem was stated and proved explicitly by
5545:
5501:
5467:
5036:
5008:
4980:
4589:
4088:
3924:
1501:
1423:
6249:
6221:, Dover Publishing Co., second revised edition, 1996.
5461:
5316:
5278:
5241:
5111:
5076:
4978:
4853:
4678:
4581:
4453:
4339:
4307:
Denoting the altitude from one side of a triangle as
4214:
4213:
4086:
3982:
3916:
3735:
3652:
3599:
3546:
3314:
3308:
for the vertices of the orthic triangle are given by
2958:
2817:
2695:
2569:
2234:
2050:
1877:
1750:
1598:
1397:
1306:
910:
735:
620:
507:
208:
6167:
http://forumgeom.fau.edu/FG2017volume17/FG201719.pdf
5754:
Clark Kimberling's Encyclopedia of Triangle Centers
4155:, this equation can also used to find the altitudes
3518:
be the line tangent to the circumcircle of triangle
483:
divides the hypotenuse into two segments of lengths
5903:
http://mathworld.wolfram.com/IsotomicConjugate.html
4326:(radius of the triangle's circumscribed circle) as
3879:The reference triangle and its orthic triangle are
3722:of the orthic and tangential triangles, are on the
2634:passing through the orthocenter of a triangle is a
6671:Animated demonstration of orthocenter construction
6297:
5965:http://mathworld.wolfram.com/JerabekHyperbola.html
5581:
5434:
5307:. The third altitude can be found by the relation
5297:
5260:
5171:
5094:
5062:
4956:
4811:
4661:
4556:Denoting the altitudes of any triangle from sides
4540:
4378:
4291:
4120:
4065:
3961:
3871:
3692:
3639:
3586:
3462:
3214:
2912:
2790:
2682:. The center of the nine-point circle lies at the
2609:{\displaystyle {\overline {HD}}={\overline {DP}}.}
2608:
2523:
2183:
1982:
1855:
1709:
1549:
1358:
1234:
889:
716:
533:
461:
130:It is common to mark the altitude with the letter
6521:Proceedings of the Edinburgh Mathematical Society
5953:http://mathworld.wolfram.com/KiepertParabola.html
3886:For more information on the orthic triangle, see
444:
440:
379:
375:
374:
341:
337:
336:
305:
301:
291:
287:
286:
6076:
5924:
4444:are the altitudes to the respective sides, then
3248:in the text) is the orthic triangle of triangle
2642:Relation to other centers, the nine-point circle
100:Altitudes can be used in the computation of the
4417:are the perpendicular distances from any point
4132:and the height is the altitude from the vertex
2540:, is extended to intersect the circumcircle at
591:Three altitudes intersecting at the orthocenter
6350:Note Whiteside's footnotes 90â92, pp. 454â456.
6043:. American Mathematical Society. p. 292.
3507:The orthic triangle is closely related to the
2550:is a chord of the circumcircle, then the foot
6100:
5987:
5975:
5742:
491:. If we denote the length of the altitude by
8:
6362:"Concurrency of the Altitudes of a Triangle"
3295:is the orthocenter of the original triangle
603:. The orthocenter lies inside the triangle
3707:, whose sides are the tangents to triangle
4177:Consider an arbitrary triangle with sides
1249:interior, on the right-angled vertex of a
724:be the side lengths. The orthocenter has
433:
412:
411:
410:
256:
255:
236:
235:
6532:
6330:
6328:
6286:
6270:
6254:
6248:
6041:Continuous symmetry: from Euclid to Klein
6030:
6028:
5869:
5867:
5829:
5827:
5544:
5500:
5466:
5462:
5460:
5421:
5412:
5401:
5392:
5381:
5376:
5367:
5356:
5351:
5342:
5331:
5326:
5317:
5315:
5283:
5277:
5246:
5240:
5161:
5152:
5141:
5132:
5121:
5112:
5110:
5075:
5054:
5035:
5026:
5007:
4998:
4979:
4977:
4945:
4930:
4920:
4905:
4895:
4880:
4870:
4855:
4852:
4792:
4787:
4762:
4757:
4732:
4727:
4709:
4694:
4680:
4677:
4643:
4638:
4622:
4617:
4601:
4596:
4588:
4580:
4524:
4514:
4508:
4497:
4487:
4481:
4470:
4460:
4454:
4452:
4353:
4344:
4338:
4277:
4268:
4257:
4248:
4237:
4228:
4215:
4212:
4110:
4102:
4087:
4085:
4002:
3996:
3987:
3981:
3962:{\displaystyle s={\tfrac {1}{2}}(a+b+c),}
3923:
3915:
3736:
3734:
3681:
3668:
3651:
3628:
3615:
3598:
3575:
3562:
3545:
3315:
3313:
3172:
3156:
3136:
3121:
3104:
3091:
3078:
3062:
3012:
2996:
2979:
2964:
2959:
2957:
2888:
2866:
2844:
2822:
2818:
2816:
2766:
2748:
2722:
2701:
2696:
2694:
2588:
2570:
2568:
2508:
2486:
2471:
2461:
2446:
2436:
2421:
2411:
2398:
2393:
2380:
2375:
2362:
2357:
2321:
2303:
2285:
2270:
2257:
2244:
2235:
2233:
2172:
2156:
2141:
2131:
2116:
2106:
2091:
2081:
2068:
2055:
2049:
1944:
1911:
1878:
1876:
1817:
1784:
1751:
1749:
1689:
1671:
1653:
1635:
1617:
1599:
1597:
1528:
1527:
1503:
1502:
1500:
1477:
1476:
1450:
1449:
1425:
1424:
1422:
1399:
1398:
1396:
1350:
1337:
1324:
1311:
1305:
1173:
1160:
1147:
1128:
1115:
1102:
1083:
1070:
1057:
1038:
1025:
1012:
993:
980:
967:
951:
938:
925:
911:
909:
736:
734:
695:
663:
631:
619:
521:
512:
506:
445:
427:
384:
368:
361:
346:
330:
323:
310:
280:
266:
257:
249:
229:
209:
207:
6360:Hajja, Mowaffaq; Martini, Horst (2013).
5627:in his (11th century) commentary on the
5233:, each of the legs is also an altitude:
3229:
93:at that vertex. It is a special case of
6346:The Mathematical Papers of Isaac Newton
6189:
6177:
6140:College Geometry / A Discovery Approach
6112:
6088:
6064:
5858:
5818:
5738:
5736:
5720:
5679:A particularly elegant proof is due to
4379:{\displaystyle h_{a}={\frac {bc}{2R}}.}
3714:'s circumcircle at its vertices; it is
2678:all lie on a single line, known as the
1359:{\displaystyle z_{H}=z_{A}+z_{B}+z_{C}}
6563:Berele, Allan; Goldman, Jerry (2001),
6035:William H. Barker, Roger Howe (2007).
3269:(does not contain a right-angle), the
1580:denote the feet of the altitudes from
1388:can be established straightforwardly:
6697:Straight lines defined for a triangle
6014:
6012:
5889:
5727:
7:
6565:Geometry: Theorems and Constructions
6344:. In Whiteside, Derek Thomas (ed.).
6341:"3.1 The 'Geometry of Curved Lines'"
6298:{\displaystyle a^{-2}+b^{-2}=d^{-2}}
6243:Voles, Roger, "Integer solutions of
6037:"§ VI.2: The classical coincidences"
5661:proved it in an unfinished treatise
3693:{\displaystyle C''=L_{C}\cap L_{A}.}
3640:{\displaystyle B''=L_{C}\cap L_{A},}
3587:{\displaystyle A''=L_{B}\cap L_{C},}
1736:of any interior point and the three
599:of the triangle, usually denoted by
6165:, Volume 17 (2017), pp. 149â156.
1497:
1419:
546:
5757:"Encyclopedia of Triangle Centers"
4972:Since the area of the triangle is
4690:
4687:
4684:
4681:
1519:
1516:
1513:
1510:
1507:
1504:
1441:
1438:
1435:
1432:
1429:
1426:
1373:, namely the altitude of triangle
534:{\displaystyle h_{c}={\sqrt {pq}}}
14:
4181:and with corresponding altitudes
3894:Some additional altitude theorems
2801:The orthocenter is closer to the
68:to a line containing the side or
6219:Challenging Problems in Geometry
6055:See also: Corollary 5.5, p. 318.
4128:where the base is taken as side
2198:as the radius of the triangle's
2029:Relation with circles and conics
5607:(proposition 5), attributed to
3511:, constructed as follows: let
1469:
409:
261:
244:
6681:Wolfram Demonstrations Project
6586:"Existence of the Orthocenter"
6499:Gauss, Carl Friedrich (1873).
6415:"XXIV. Geometry and geometers"
6369:Mathematische Semesterberichte
5221:In a right triangle with legs
4801:
4774:
4771:
4744:
4741:
4714:
4049:
4037:
4034:
4022:
4019:
4007:
3953:
3935:
3899:Altitude in terms of the sides
3110:
3071:
2534:If any altitude, for example,
2505:
2495:
1538:
1487:
1460:
1409:
1179:
1137:
1134:
1095:
1089:
1047:
1044:
1005:
999:
960:
957:
918:
226:
213:
1:
6625:(5th ed.), Brooks/Cole,
5835:""Orthocenter of a triangle""
5667:
5620:
6142:, HarperCollins, p. 6,
4940:
4915:
4890:
4865:
4831:is any point on an altitude
4823:General point on an altitude
4076:This follows from combining
3903:For any triangle with sides
3700:The tangential triangle is
3131:
2974:
2898:
2876:
2854:
2832:
2776:
2758:
2732:
2711:
2598:
2580:
2481:
2456:
2431:
2331:
2313:
2295:
2151:
2126:
2101:
1968:
1955:
1935:
1922:
1902:
1889:
1841:
1828:
1808:
1795:
1775:
1762:
1699:
1681:
1663:
1645:
1627:
1609:
1369:is represented by the point
705:
673:
641:
498:, we then have the relation
476:, the altitude drawn to the
149:on the 3 triangles of sides
6605:Advanced Euclidean Geometry
6603:Johnson, Roger A. (2007) ,
6022:91, November 2007, 436â452.
5447:inverse Pythagorean theorem
4330:, the altitude is given by
2024:or orthocentric quadrangle.
551:inverse Pythagorean theorem
6713:
6667:With interactive animation
6387:Hogendijk, Jan P. (2008).
5601:texts, but is used in the
5445:This is also known as the
5070:, the triangle inequality
2645:
2015:anticomplementary triangle
2009:of the orthocenter is the
1998:of the orthocenter is the
580:
18:
6673:Compass and straightedge.
6665:Orthocenter of a triangle
6534:10.1017/S0013091500036762
6436:10.1080/14786445008646583
6381:10.1007/s00591-013-0123-z
6101:Berele & Goldman 2001
5988:Berele & Goldman 2001
5976:Berele & Goldman 2001
5743:Berele & Goldman 2001
5683:(1804) and independently
4314:, the other two sides as
6621:Smart, James R. (1998),
6552:Altshiller-Court, Nathan
6474:Servois, Francois-Joseph
6234:89 (November 2005), 494.
6217:and Charles T. Salkind,
5792:, page 90, Proposition 3
5663:Geometry of Curved Lines
5095:{\displaystyle a<b+c}
4196:. The altitudes and the
6441:Footnote on pp. 207â208
6411:Davies, Thomas Stephens
6322:92, July 2008, 313â317.
6309:83, July 1999, 269â271.
6205:89, November 2005, 494.
6128:82, July 1998, 298-299.
5693:perpendicular bisectors
5681:François-Joseph Servois
5617:Mathematical Collection
5298:{\displaystyle h_{b}=a}
5261:{\displaystyle h_{a}=b}
3973:(the base) is given by
3969:the altitude from side
1719:The circle centered at
901:barycentric coordinates
106:trigonometric functions
72:opposite the apex (the
34:is inside the triangle.
6493:Geometrie der Stellung
6423:Philosophical Magazine
6299:
6138:Kay, David C. (1993),
6115:, p. 172, Section 270c
5583:
5436:
5299:
5262:
5218:
5173:
5096:
5064:
4958:
4813:
4663:
4542:
4380:
4293:
4122:
4067:
3963:
3873:
3694:
3641:
3588:
3464:
3255:
3216:
2923:In terms of the sides
2914:
2792:
2610:
2525:
2194:In addition, denoting
2185:
1984:
1857:
1711:
1566:James Joseph Sylvester
1551:
1360:
1236:
891:
718:
592:
562:
543:Geometric mean theorem
535:
469:
463:
35:
6513:Mackay, John Sturgeon
6489:Gauss, Carl Friedrich
6300:
6215:Alfred S. Posamentier
6192:, p. 74, Section 103c
6180:, p. 71, Section 101a
6091:, p. 168, Section 264
6077:Altshiller-Court 2007
6067:, p. 199, Section 315
5925:Altshiller-Court 2007
5878:University of Georgia
5861:, p. 176, Section 278
5821:, p. 163, Section 255
5584:
5437:
5300:
5263:
5216:
5174:
5097:
5065:
4959:
4814:
4664:
4543:
4381:
4322:, and the triangle's
4294:
4123:
4068:
3964:
3874:
3695:
3642:
3589:
3465:
3306:Trilinear coordinates
3233:
3217:
2915:
2793:
2636:rectangular hyperbola
2611:
2526:
2186:
1985:
1858:
1734:more general property
1712:
1552:
1361:
1253:, and exterior to an
1237:
892:
726:trilinear coordinates
719:
590:
560:
536:
464:
144:
127:of the vertex angle.
115:(a triangle with two
95:orthogonal projection
91:dropping the altitude
29:
6582:Bogomolny, Alexander
6446:Bogomolny, Alexander
6320:Mathematical Gazette
6307:Mathematical Gazette
6247:
6232:Mathematical Gazette
6203:Mathematical Gazette
6126:Mathematical Gazette
6020:Mathematical Gazette
5685:Carl Friedrich Gauss
5631:, and attributed to
5459:
5452:Note in particular:
5314:
5276:
5239:
5199:equilateral triangle
5189:Equilateral triangle
5109:
5074:
4976:
4851:
4676:
4579:
4451:
4337:
4303:Circumradius theorem
4211:
4084:
3980:
3914:
3881:orthologic triangles
3733:
3720:center of similitude
3650:
3597:
3544:
3312:
2956:
2815:
2693:
2567:
2232:
2217:as the radii of its
2048:
1875:
1748:
1596:
1584:respectively. Then:
1562:problem of Sylvester
1395:
1304:
1286:and assume that the
908:
733:
618:
505:
206:
6679:by Jay Warendorff,
6491:(1810). "ZusÀtze".
6163:Forum Geometricorum
6003:14 (2014), 51-61.
6001:Forum Geometricorum
5939:Forum Geometricorum
5676:proved it in 1749.
5640: 10th century
5386:
5361:
5336:
4968:Triangle inequality
4800:
4770:
4740:
4651:
4630:
4609:
3509:tangential triangle
2403:
2385:
2367:
2037:of the triangle by
2022:orthocentric system
583:Orthocentric system
147:Pythagoras' theorem
21:Orthocentric system
6648:Weisstein, Eric W.
6295:
5599:Greek mathematical
5579:
5577:
5573:
5510:
5476:
5432:
5372:
5347:
5322:
5295:
5258:
5219:
5169:
5092:
5060:
5045:
5017:
4989:
4954:
4809:
4783:
4753:
4723:
4659:
4657:
4634:
4613:
4592:
4538:
4421:to the sides, and
4376:
4289:
4288:
4118:
4097:
4063:
3959:
3933:
3869:
3867:
3690:
3637:
3584:
3540:analogously. Let
3460:
3458:
3256:
3212:
3210:
2910:
2908:
2788:
2786:
2606:
2521:
2519:
2389:
2371:
2353:
2181:
2007:isotomic conjugate
1996:isogonal conjugate
1980:
1853:
1707:
1547:
1526:
1524:
1448:
1446:
1356:
1264:, let the points
1232:
1230:
887:
885:
714:
593:
563:
531:
470:
459:
457:
113:isosceles triangle
102:area of a triangle
36:
6677:Fagnano's Problem
6623:Modern Geometries
6614:978-0-486-46237-0
6567:, Prentice Hall,
6050:978-0-8218-3900-3
5941:6, 2006, 335â342.
5790:978-0-8176-4326-3
5780:Andreescu, Titu;
5710:Median (geometry)
5653:proved it in his
5572:
5509:
5475:
5427:
5407:
5387:
5362:
5337:
5203:Viviani's theorem
5167:
5147:
5127:
5044:
5016:
4988:
4943:
4918:
4893:
4868:
4804:
4656:
4530:
4503:
4476:
4371:
4283:
4263:
4243:
4223:
4173:Inradius theorems
4113:
4105:
4096:
4058:
4052:
3932:
3489:Fagnano's problem
3279:altitude triangle
3134:
2977:
2901:
2879:
2857:
2835:
2779:
2761:
2735:
2714:
2676:nine-point circle
2670:, and the center
2648:Nine-point circle
2601:
2583:
2484:
2459:
2434:
2334:
2316:
2298:
2154:
2129:
2104:
1972:
1971:
1958:
1939:
1938:
1925:
1906:
1905:
1892:
1845:
1844:
1831:
1812:
1811:
1798:
1779:
1778:
1765:
1702:
1684:
1666:
1648:
1630:
1612:
1541:
1496:
1490:
1463:
1418:
1412:
708:
676:
644:
529:
453:
394:
356:
6704:
6661:
6660:
6635:
6617:
6599:
6597:
6596:
6577:
6559:
6556:College Geometry
6539:
6538:
6536:
6508:
6496:
6485:
6470:
6464:
6463:
6461:
6460:
6439:
6430:(249): 198â212.
6419:
6407:
6401:
6400:
6384:
6366:
6357:
6351:
6349:
6343:
6332:
6323:
6316:
6310:
6304:
6302:
6301:
6296:
6294:
6293:
6278:
6277:
6262:
6261:
6241:
6235:
6228:
6222:
6212:
6206:
6199:
6193:
6187:
6181:
6175:
6169:
6159:
6153:
6152:
6135:
6129:
6122:
6116:
6110:
6104:
6098:
6092:
6086:
6080:
6074:
6068:
6062:
6056:
6054:
6032:
6023:
6016:
6007:
5997:
5991:
5985:
5979:
5973:
5967:
5961:
5955:
5949:
5943:
5934:
5928:
5922:
5916:
5911:
5905:
5899:
5893:
5887:
5881:
5871:
5862:
5856:
5850:
5849:
5847:
5846:
5837:. Archived from
5831:
5822:
5816:
5810:
5799:
5793:
5778:
5772:
5771:
5769:
5768:
5759:. Archived from
5752:
5746:
5740:
5731:
5725:
5671:
5669:
5641:
5638:
5622:
5588:
5586:
5585:
5580:
5578:
5574:
5571:
5563:
5546:
5511:
5502:
5477:
5468:
5441:
5439:
5438:
5433:
5428:
5426:
5425:
5413:
5408:
5406:
5405:
5393:
5388:
5385:
5380:
5368:
5363:
5360:
5355:
5343:
5338:
5335:
5330:
5318:
5306:
5304:
5302:
5301:
5296:
5288:
5287:
5269:
5267:
5265:
5264:
5259:
5251:
5250:
5232:
5228:
5224:
5196:
5178:
5176:
5175:
5170:
5168:
5166:
5165:
5153:
5148:
5146:
5145:
5133:
5128:
5126:
5125:
5113:
5101:
5099:
5098:
5093:
5069:
5067:
5066:
5061:
5059:
5058:
5046:
5037:
5031:
5030:
5018:
5009:
5003:
5002:
4990:
4981:
4963:
4961:
4960:
4955:
4950:
4949:
4944:
4939:
4931:
4925:
4924:
4919:
4914:
4906:
4900:
4899:
4894:
4889:
4881:
4875:
4874:
4869:
4864:
4856:
4843:
4837:of any triangle
4836:
4835:
4830:
4818:
4816:
4815:
4810:
4805:
4799:
4791:
4769:
4761:
4739:
4731:
4710:
4702:
4701:
4693:
4668:
4666:
4665:
4660:
4658:
4652:
4650:
4642:
4629:
4621:
4608:
4600:
4590:
4574:
4560:respectively as
4559:
4547:
4545:
4544:
4539:
4531:
4529:
4528:
4519:
4518:
4509:
4504:
4502:
4501:
4492:
4491:
4482:
4477:
4475:
4474:
4465:
4464:
4455:
4443:
4420:
4416:
4385:
4383:
4382:
4377:
4372:
4370:
4362:
4354:
4349:
4348:
4329:
4321:
4317:
4313:
4298:
4296:
4295:
4290:
4284:
4282:
4281:
4269:
4264:
4262:
4261:
4249:
4244:
4242:
4241:
4229:
4224:
4216:
4203:
4195:
4180:
4169:, respectively.
4168:
4161:
4154:
4150:
4146:
4139:
4135:
4131:
4127:
4125:
4124:
4119:
4114:
4111:
4106:
4103:
4098:
4089:
4072:
4070:
4069:
4064:
4059:
4054:
4053:
4003:
3997:
3992:
3991:
3972:
3968:
3966:
3965:
3960:
3934:
3925:
3906:
3878:
3876:
3875:
3870:
3868:
3833:
3790:
3747:
3713:
3706:
3699:
3697:
3696:
3691:
3686:
3685:
3673:
3672:
3660:
3646:
3644:
3643:
3638:
3633:
3632:
3620:
3619:
3607:
3593:
3591:
3590:
3585:
3580:
3579:
3567:
3566:
3554:
3539:
3528:
3524:
3517:
3503:
3478:collinear points
3469:
3467:
3466:
3461:
3459:
3301:
3294:
3287:
3264:
3258:If the triangle
3254:
3247:
3240:
3221:
3219:
3218:
3213:
3211:
3177:
3176:
3161:
3160:
3141:
3140:
3135:
3130:
3122:
3109:
3108:
3096:
3095:
3083:
3082:
3067:
3066:
3048:
3017:
3016:
3001:
3000:
2984:
2983:
2978:
2973:
2965:
2948:
2941:
2934:
2930:
2926:
2919:
2917:
2916:
2911:
2909:
2902:
2897:
2889:
2880:
2875:
2867:
2858:
2853:
2845:
2836:
2831:
2823:
2807:
2797:
2795:
2794:
2789:
2787:
2780:
2775:
2767:
2762:
2757:
2749:
2743:
2736:
2731:
2723:
2715:
2710:
2702:
2699:
2673:
2669:
2662:
2655:
2652:The orthocenter
2615:
2613:
2612:
2607:
2602:
2597:
2589:
2584:
2579:
2571:
2559:
2558:
2554:bisects segment
2553:
2549:
2548:
2543:
2539:
2538:
2530:
2528:
2527:
2522:
2520:
2513:
2512:
2491:
2490:
2485:
2480:
2472:
2466:
2465:
2460:
2455:
2447:
2441:
2440:
2435:
2430:
2422:
2416:
2415:
2402:
2397:
2384:
2379:
2366:
2361:
2351:
2335:
2330:
2322:
2317:
2312:
2304:
2299:
2294:
2286:
2275:
2274:
2262:
2261:
2249:
2248:
2238:
2224:
2216:
2197:
2190:
2188:
2187:
2182:
2177:
2176:
2161:
2160:
2155:
2150:
2142:
2136:
2135:
2130:
2125:
2117:
2111:
2110:
2105:
2100:
2092:
2086:
2085:
2073:
2072:
2060:
2059:
2040:
2002:of the triangle.
1989:
1987:
1986:
1981:
1973:
1967:
1959:
1954:
1946:
1945:
1940:
1934:
1926:
1921:
1913:
1912:
1907:
1901:
1893:
1888:
1880:
1879:
1862:
1860:
1859:
1854:
1846:
1840:
1832:
1827:
1819:
1818:
1813:
1807:
1799:
1794:
1786:
1785:
1780:
1774:
1766:
1761:
1753:
1752:
1722:
1716:
1714:
1713:
1708:
1703:
1698:
1690:
1685:
1680:
1672:
1667:
1662:
1654:
1649:
1644:
1636:
1631:
1626:
1618:
1613:
1608:
1600:
1583:
1579:
1556:
1554:
1553:
1548:
1543:
1542:
1537:
1529:
1525:
1523:
1522:
1492:
1491:
1486:
1478:
1465:
1464:
1459:
1451:
1447:
1445:
1444:
1414:
1413:
1408:
1400:
1383:
1379:
1372:
1365:
1363:
1362:
1357:
1355:
1354:
1342:
1341:
1329:
1328:
1316:
1315:
1296:
1285:
1267:
1247:acute triangle's
1241:
1239:
1238:
1233:
1231:
1185:
1178:
1177:
1165:
1164:
1152:
1151:
1133:
1132:
1120:
1119:
1107:
1106:
1088:
1087:
1075:
1074:
1062:
1061:
1043:
1042:
1030:
1029:
1017:
1016:
998:
997:
985:
984:
972:
971:
956:
955:
943:
942:
930:
929:
914:
896:
894:
893:
888:
886:
777:
739:
723:
721:
720:
715:
713:
709:
704:
696:
681:
677:
672:
664:
649:
645:
640:
632:
613:
602:
540:
538:
537:
532:
530:
522:
517:
516:
497:
490:
486:
482:
468:
466:
465:
460:
458:
454:
446:
432:
431:
395:
390:
389:
388:
373:
372:
362:
357:
352:
351:
350:
335:
334:
324:
315:
314:
285:
284:
271:
270:
254:
253:
234:
233:
200:
184:
168:
133:
54:through a given
6712:
6711:
6707:
6706:
6705:
6703:
6702:
6701:
6687:
6686:
6646:
6645:
6642:
6633:
6620:
6615:
6602:
6594:
6592:
6580:
6575:
6562:
6550:
6547:
6542:
6511:
6509:
6498:
6497:republished in
6487:
6486:
6472:
6471:
6467:
6458:
6456:
6444:
6417:
6409:
6408:
6404:
6386:
6385:
6364:
6359:
6358:
6354:
6334:
6333:
6326:
6317:
6313:
6282:
6266:
6250:
6245:
6244:
6242:
6238:
6229:
6225:
6213:
6209:
6200:
6196:
6188:
6184:
6176:
6172:
6160:
6156:
6150:
6137:
6136:
6132:
6123:
6119:
6111:
6107:
6099:
6095:
6087:
6083:
6075:
6071:
6063:
6059:
6051:
6034:
6033:
6026:
6017:
6010:
5998:
5994:
5986:
5982:
5974:
5970:
5962:
5958:
5950:
5946:
5935:
5931:
5923:
5919:
5912:
5908:
5900:
5896:
5888:
5884:
5872:
5865:
5857:
5853:
5844:
5842:
5833:
5832:
5825:
5817:
5813:
5800:
5796:
5779:
5775:
5766:
5764:
5755:
5753:
5749:
5741:
5734:
5726:
5722:
5718:
5705:Triangle center
5701:
5689:medial triangle
5674:William Chapple
5665:
5651:Samuel Marolois
5639:
5595:
5576:
5575:
5564:
5547:
5537:
5528:
5527:
5493:
5457:
5456:
5417:
5397:
5312:
5311:
5279:
5274:
5273:
5271:
5242:
5237:
5236:
5234:
5230:
5229:and hypotenuse
5226:
5222:
5211:
5194:
5193:From any point
5191:
5186:
5157:
5137:
5117:
5107:
5106:
5072:
5071:
5050:
5022:
4994:
4974:
4973:
4970:
4932:
4929:
4907:
4904:
4882:
4879:
4857:
4854:
4849:
4848:
4838:
4833:
4832:
4828:
4825:
4679:
4674:
4673:
4591:
4577:
4576:
4573:
4569:
4565:
4561:
4557:
4554:
4520:
4510:
4493:
4483:
4466:
4456:
4449:
4448:
4442:
4435:
4428:
4422:
4418:
4415:
4408:
4401:
4395:
4392:
4363:
4355:
4340:
4335:
4334:
4327:
4319:
4315:
4312:
4308:
4305:
4273:
4253:
4233:
4209:
4208:
4204:are related by
4201:
4194:
4190:
4186:
4182:
4178:
4175:
4167:
4163:
4160:
4156:
4152:
4148:
4144:
4137:
4136:(opposite side
4133:
4129:
4082:
4081:
4078:Heron's formula
3998:
3983:
3978:
3977:
3970:
3912:
3911:
3904:
3901:
3896:
3866:
3865:
3857:
3852:
3847:
3842:
3837:
3826:
3823:
3822:
3817:
3812:
3804:
3799:
3794:
3783:
3780:
3779:
3774:
3769:
3764:
3759:
3751:
3740:
3731:
3730:
3708:
3701:
3677:
3664:
3653:
3648:
3647:
3624:
3611:
3600:
3595:
3594:
3571:
3558:
3547:
3542:
3541:
3538:
3534:
3530:
3526:
3519:
3516:
3512:
3498:
3457:
3456:
3451:
3446:
3435:
3430:
3419:
3410:
3409:
3398:
3393:
3388:
3383:
3372:
3363:
3362:
3351:
3346:
3335:
3330:
3325:
3310:
3309:
3296:
3289:
3282:
3275:orthic triangle
3259:
3249:
3242:
3241:(respectively,
3235:
3228:
3226:Orthic triangle
3209:
3208:
3168:
3152:
3142:
3123:
3120:
3117:
3116:
3100:
3087:
3074:
3058:
3046:
3045:
3008:
2992:
2985:
2966:
2963:
2954:
2953:
2946:
2939:
2932:
2928:
2924:
2907:
2906:
2890:
2881:
2868:
2863:
2862:
2846:
2837:
2824:
2813:
2812:
2805:
2785:
2784:
2768:
2750:
2741:
2740:
2724:
2703:
2691:
2690:
2671:
2667:
2660:
2653:
2650:
2644:
2590:
2572:
2565:
2564:
2556:
2555:
2551:
2546:
2545:
2541:
2536:
2535:
2518:
2517:
2504:
2473:
2470:
2448:
2445:
2423:
2420:
2407:
2349:
2348:
2323:
2305:
2287:
2266:
2253:
2240:
2230:
2229:
2222:
2215:
2211:
2207:
2203:
2195:
2168:
2143:
2140:
2118:
2115:
2093:
2090:
2077:
2064:
2051:
2046:
2045:
2038:
2031:
2011:symmedian point
1960:
1947:
1927:
1914:
1894:
1881:
1873:
1872:
1833:
1820:
1800:
1787:
1767:
1754:
1746:
1745:
1720:
1691:
1673:
1655:
1637:
1619:
1601:
1594:
1593:
1581:
1577:
1574:
1530:
1479:
1452:
1401:
1393:
1392:
1381:
1374:
1370:
1346:
1333:
1320:
1307:
1302:
1301:
1291:
1284:
1280:
1276:
1272:
1265:
1255:obtuse triangle
1229:
1228:
1183:
1182:
1169:
1156:
1143:
1124:
1111:
1098:
1079:
1066:
1053:
1034:
1021:
1008:
989:
976:
963:
947:
934:
921:
906:
905:
884:
883:
775:
774:
731:
730:
697:
691:
665:
659:
633:
627:
616:
615:
611:
600:
585:
579:
508:
503:
502:
496:
492:
488:
484:
480:
456:
455:
423:
413:
397:
396:
380:
364:
363:
342:
326:
325:
316:
306:
276:
273:
272:
262:
245:
237:
225:
204:
203:
202:
186:
170:
155: +
150:
131:
24:
17:
12:
11:
5:
6710:
6708:
6700:
6699:
6689:
6688:
6685:
6684:
6674:
6668:
6662:
6641:
6640:External links
6638:
6637:
6636:
6631:
6618:
6613:
6600:
6578:
6573:
6560:
6546:
6543:
6541:
6540:
6465:
6402:
6375:(2): 249â260.
6352:
6324:
6311:
6292:
6289:
6285:
6281:
6276:
6273:
6269:
6265:
6260:
6257:
6253:
6236:
6223:
6207:
6194:
6182:
6170:
6154:
6148:
6130:
6117:
6105:
6093:
6081:
6069:
6057:
6049:
6024:
6008:
5992:
5980:
5968:
5956:
5944:
5929:
5917:
5906:
5894:
5882:
5863:
5851:
5823:
5811:
5794:
5782:Andrica, Dorin
5773:
5747:
5732:
5719:
5717:
5714:
5713:
5712:
5707:
5700:
5697:
5647:Book of Lemmas
5629:Book of Lemmas
5604:Book of Lemmas
5594:
5591:
5590:
5589:
5570:
5567:
5562:
5559:
5556:
5553:
5550:
5543:
5540:
5538:
5536:
5533:
5530:
5529:
5526:
5523:
5520:
5517:
5514:
5508:
5505:
5499:
5496:
5494:
5492:
5489:
5486:
5483:
5480:
5474:
5471:
5465:
5464:
5443:
5442:
5431:
5424:
5420:
5416:
5411:
5404:
5400:
5396:
5391:
5384:
5379:
5375:
5371:
5366:
5359:
5354:
5350:
5346:
5341:
5334:
5329:
5325:
5321:
5294:
5291:
5286:
5282:
5257:
5254:
5249:
5245:
5210:
5209:Right triangle
5207:
5190:
5187:
5185:
5182:
5181:
5180:
5164:
5160:
5156:
5151:
5144:
5140:
5136:
5131:
5124:
5120:
5116:
5091:
5088:
5085:
5082:
5079:
5057:
5053:
5049:
5043:
5040:
5034:
5029:
5025:
5021:
5015:
5012:
5006:
5001:
4997:
4993:
4987:
4984:
4969:
4966:
4965:
4964:
4953:
4948:
4942:
4938:
4935:
4928:
4923:
4917:
4913:
4910:
4903:
4898:
4892:
4888:
4885:
4878:
4873:
4867:
4863:
4860:
4824:
4821:
4820:
4819:
4808:
4803:
4798:
4795:
4790:
4786:
4782:
4779:
4776:
4773:
4768:
4765:
4760:
4756:
4752:
4749:
4746:
4743:
4738:
4735:
4730:
4726:
4722:
4719:
4716:
4713:
4708:
4705:
4700:
4697:
4692:
4689:
4686:
4683:
4655:
4649:
4646:
4641:
4637:
4633:
4628:
4625:
4620:
4616:
4612:
4607:
4604:
4599:
4595:
4587:
4584:
4571:
4567:
4563:
4553:
4550:
4549:
4548:
4537:
4534:
4527:
4523:
4517:
4513:
4507:
4500:
4496:
4490:
4486:
4480:
4473:
4469:
4463:
4459:
4440:
4433:
4426:
4413:
4406:
4399:
4391:
4390:Interior point
4388:
4387:
4386:
4375:
4369:
4366:
4361:
4358:
4352:
4347:
4343:
4310:
4304:
4301:
4300:
4299:
4287:
4280:
4276:
4272:
4267:
4260:
4256:
4252:
4247:
4240:
4236:
4232:
4227:
4222:
4219:
4192:
4188:
4184:
4174:
4171:
4165:
4158:
4143:By exchanging
4117:
4109:
4101:
4095:
4092:
4074:
4073:
4062:
4057:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4001:
3995:
3990:
3986:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3931:
3928:
3922:
3919:
3900:
3897:
3895:
3892:
3864:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3836:
3832:
3829:
3825:
3824:
3821:
3818:
3816:
3813:
3811:
3808:
3805:
3803:
3800:
3798:
3795:
3793:
3789:
3786:
3782:
3781:
3778:
3775:
3773:
3770:
3768:
3765:
3763:
3760:
3758:
3755:
3752:
3750:
3746:
3743:
3739:
3738:
3689:
3684:
3680:
3676:
3671:
3667:
3663:
3659:
3656:
3636:
3631:
3627:
3623:
3618:
3614:
3610:
3606:
3603:
3583:
3578:
3574:
3570:
3565:
3561:
3557:
3553:
3550:
3536:
3532:
3514:
3485:acute triangle
3474:extended sides
3455:
3452:
3450:
3447:
3445:
3442:
3439:
3436:
3434:
3431:
3429:
3426:
3423:
3420:
3418:
3415:
3412:
3411:
3408:
3405:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3376:
3373:
3371:
3368:
3365:
3364:
3361:
3358:
3355:
3352:
3350:
3347:
3345:
3342:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3318:
3317:
3271:pedal triangle
3227:
3224:
3223:
3222:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3175:
3171:
3167:
3164:
3159:
3155:
3151:
3148:
3145:
3143:
3139:
3133:
3129:
3126:
3119:
3118:
3115:
3112:
3107:
3103:
3099:
3094:
3090:
3086:
3081:
3077:
3073:
3070:
3065:
3061:
3057:
3054:
3051:
3049:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3015:
3011:
3007:
3004:
2999:
2995:
2991:
2988:
2986:
2982:
2976:
2972:
2969:
2962:
2961:
2921:
2920:
2905:
2900:
2896:
2893:
2887:
2884:
2882:
2878:
2874:
2871:
2865:
2864:
2861:
2856:
2852:
2849:
2843:
2840:
2838:
2834:
2830:
2827:
2821:
2820:
2799:
2798:
2783:
2778:
2774:
2771:
2765:
2760:
2756:
2753:
2747:
2744:
2742:
2739:
2734:
2730:
2727:
2721:
2718:
2713:
2709:
2706:
2700:
2698:
2646:Main article:
2643:
2640:
2617:
2616:
2605:
2600:
2596:
2593:
2587:
2582:
2578:
2575:
2532:
2531:
2516:
2511:
2507:
2503:
2500:
2497:
2494:
2489:
2483:
2479:
2476:
2469:
2464:
2458:
2454:
2451:
2444:
2439:
2433:
2429:
2426:
2419:
2414:
2410:
2406:
2401:
2396:
2392:
2388:
2383:
2378:
2374:
2370:
2365:
2360:
2356:
2352:
2350:
2347:
2344:
2341:
2338:
2333:
2329:
2326:
2320:
2315:
2311:
2308:
2302:
2297:
2293:
2290:
2284:
2281:
2278:
2273:
2269:
2265:
2260:
2256:
2252:
2247:
2243:
2239:
2237:
2213:
2209:
2205:
2192:
2191:
2180:
2175:
2171:
2167:
2164:
2159:
2153:
2149:
2146:
2139:
2134:
2128:
2124:
2121:
2114:
2109:
2103:
2099:
2096:
2089:
2084:
2080:
2076:
2071:
2067:
2063:
2058:
2054:
2030:
2027:
2026:
2025:
2018:
2003:
1991:
1990:
1979:
1976:
1970:
1966:
1963:
1957:
1953:
1950:
1943:
1937:
1933:
1930:
1924:
1920:
1917:
1910:
1904:
1900:
1897:
1891:
1887:
1884:
1869:
1868:
1864:
1863:
1852:
1849:
1843:
1839:
1836:
1830:
1826:
1823:
1816:
1810:
1806:
1803:
1797:
1793:
1790:
1783:
1777:
1773:
1770:
1764:
1760:
1757:
1742:
1741:
1729:
1728:
1717:
1706:
1701:
1697:
1694:
1688:
1683:
1679:
1676:
1670:
1665:
1661:
1658:
1652:
1647:
1643:
1640:
1634:
1629:
1625:
1622:
1616:
1611:
1607:
1604:
1590:
1589:
1573:
1570:
1564:, proposed by
1558:
1557:
1546:
1540:
1536:
1533:
1521:
1518:
1515:
1512:
1509:
1506:
1499:
1495:
1489:
1485:
1482:
1475:
1472:
1468:
1462:
1458:
1455:
1443:
1440:
1437:
1434:
1431:
1428:
1421:
1417:
1411:
1407:
1404:
1367:
1366:
1353:
1349:
1345:
1340:
1336:
1332:
1327:
1323:
1319:
1314:
1310:
1282:
1278:
1274:
1268:represent the
1251:right triangle
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1186:
1184:
1181:
1176:
1172:
1168:
1163:
1159:
1155:
1150:
1146:
1142:
1139:
1136:
1131:
1127:
1123:
1118:
1114:
1110:
1105:
1101:
1097:
1094:
1091:
1086:
1082:
1078:
1073:
1069:
1065:
1060:
1056:
1052:
1049:
1046:
1041:
1037:
1033:
1028:
1024:
1020:
1015:
1011:
1007:
1004:
1001:
996:
992:
988:
983:
979:
975:
970:
966:
962:
959:
954:
950:
946:
941:
937:
933:
928:
924:
920:
917:
915:
913:
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:
789:
786:
783:
780:
778:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
738:
712:
707:
703:
700:
694:
690:
687:
684:
680:
675:
671:
668:
662:
658:
655:
652:
648:
643:
639:
636:
630:
626:
623:
605:if and only if
578:
575:
555:
554:
528:
525:
520:
515:
511:
494:
474:right triangle
452:
449:
443:
439:
436:
430:
426:
422:
419:
416:
414:
408:
405:
402:
399:
398:
393:
387:
383:
378:
371:
367:
360:
355:
349:
345:
340:
333:
329:
322:
319:
317:
313:
309:
304:
300:
297:
294:
290:
283:
279:
275:
274:
269:
265:
260:
252:
248:
243:
240:
238:
232:
228:
224:
221:
218:
215:
212:
211:
125:angle bisector
32:acute triangle
15:
13:
10:
9:
6:
4:
3:
2:
6709:
6698:
6695:
6694:
6692:
6682:
6678:
6675:
6672:
6669:
6666:
6663:
6658:
6657:
6652:
6649:
6644:
6643:
6639:
6634:
6632:0-534-35188-3
6628:
6624:
6619:
6616:
6610:
6606:
6601:
6591:
6587:
6583:
6579:
6576:
6574:0-13-087121-4
6570:
6566:
6561:
6557:
6553:
6549:
6548:
6544:
6535:
6530:
6526:
6522:
6518:
6514:
6506:
6502:
6494:
6490:
6483:
6479:
6475:
6469:
6466:
6455:
6451:
6447:
6442:
6437:
6433:
6429:
6425:
6424:
6416:
6412:
6406:
6403:
6398:
6394:
6390:
6382:
6378:
6374:
6370:
6363:
6356:
6353:
6347:
6342:
6337:
6336:Newton, Isaac
6331:
6329:
6325:
6321:
6315:
6312:
6308:
6290:
6287:
6283:
6279:
6274:
6271:
6267:
6263:
6258:
6255:
6251:
6240:
6237:
6233:
6227:
6224:
6220:
6216:
6211:
6208:
6204:
6198:
6195:
6191:
6186:
6183:
6179:
6174:
6171:
6168:
6164:
6158:
6155:
6151:
6149:0-06-500006-4
6145:
6141:
6134:
6131:
6127:
6121:
6118:
6114:
6109:
6106:
6103:, pp. 120-122
6102:
6097:
6094:
6090:
6085:
6082:
6078:
6073:
6070:
6066:
6061:
6058:
6052:
6046:
6042:
6038:
6031:
6029:
6025:
6021:
6015:
6013:
6009:
6006:
6002:
5996:
5993:
5990:, pp. 124-126
5989:
5984:
5981:
5977:
5972:
5969:
5966:
5960:
5957:
5954:
5948:
5945:
5942:
5940:
5933:
5930:
5926:
5921:
5918:
5915:
5910:
5907:
5904:
5898:
5895:
5891:
5886:
5883:
5879:
5875:
5870:
5868:
5864:
5860:
5855:
5852:
5841:on 2012-07-05
5840:
5836:
5830:
5828:
5824:
5820:
5815:
5812:
5808:
5807:0-486-61348-8
5804:
5798:
5795:
5791:
5787:
5783:
5777:
5774:
5763:on 2012-04-19
5762:
5758:
5751:
5748:
5744:
5739:
5737:
5733:
5729:
5724:
5721:
5715:
5711:
5708:
5706:
5703:
5702:
5698:
5696:
5694:
5690:
5686:
5682:
5677:
5675:
5664:
5660:
5656:
5652:
5648:
5643:
5634:
5630:
5626:
5618:
5614:
5610:
5606:
5605:
5600:
5592:
5568:
5565:
5560:
5557:
5554:
5551:
5548:
5541:
5539:
5534:
5531:
5524:
5521:
5518:
5515:
5512:
5506:
5503:
5497:
5495:
5490:
5487:
5484:
5481:
5478:
5472:
5469:
5455:
5454:
5453:
5450:
5448:
5429:
5422:
5418:
5414:
5409:
5402:
5398:
5394:
5389:
5382:
5377:
5373:
5369:
5364:
5357:
5352:
5348:
5344:
5339:
5332:
5327:
5323:
5319:
5310:
5309:
5308:
5292:
5289:
5284:
5280:
5255:
5252:
5247:
5243:
5215:
5208:
5206:
5204:
5200:
5188:
5184:Special cases
5183:
5162:
5158:
5154:
5149:
5142:
5138:
5134:
5129:
5122:
5118:
5114:
5105:
5104:
5103:
5089:
5086:
5083:
5080:
5077:
5055:
5051:
5047:
5041:
5038:
5032:
5027:
5023:
5019:
5013:
5010:
5004:
4999:
4995:
4991:
4985:
4982:
4967:
4951:
4946:
4936:
4933:
4926:
4921:
4911:
4908:
4901:
4896:
4886:
4883:
4876:
4871:
4861:
4858:
4847:
4846:
4845:
4842:
4822:
4806:
4796:
4793:
4788:
4784:
4780:
4777:
4766:
4763:
4758:
4754:
4750:
4747:
4736:
4733:
4728:
4724:
4720:
4717:
4711:
4706:
4703:
4698:
4695:
4672:
4671:
4670:
4653:
4647:
4644:
4639:
4635:
4631:
4626:
4623:
4618:
4614:
4610:
4605:
4602:
4597:
4593:
4585:
4582:
4551:
4535:
4532:
4525:
4521:
4515:
4511:
4505:
4498:
4494:
4488:
4484:
4478:
4471:
4467:
4461:
4457:
4447:
4446:
4445:
4439:
4432:
4425:
4412:
4405:
4398:
4389:
4373:
4367:
4364:
4359:
4356:
4350:
4345:
4341:
4333:
4332:
4331:
4325:
4302:
4285:
4278:
4274:
4270:
4265:
4258:
4254:
4250:
4245:
4238:
4234:
4230:
4225:
4220:
4217:
4207:
4206:
4205:
4199:
4172:
4170:
4141:
4115:
4107:
4099:
4093:
4090:
4079:
4060:
4055:
4046:
4043:
4040:
4031:
4028:
4025:
4016:
4013:
4010:
4004:
3999:
3993:
3988:
3984:
3976:
3975:
3974:
3956:
3950:
3947:
3944:
3941:
3938:
3929:
3926:
3920:
3917:
3910:
3909:semiperimeter
3898:
3893:
3891:
3889:
3884:
3882:
3862:
3859:
3854:
3849:
3844:
3839:
3834:
3830:
3827:
3819:
3814:
3809:
3806:
3801:
3796:
3791:
3787:
3784:
3776:
3771:
3766:
3761:
3756:
3753:
3748:
3744:
3741:
3727:
3725:
3721:
3717:
3712:
3705:
3687:
3682:
3678:
3674:
3669:
3665:
3661:
3657:
3654:
3634:
3629:
3625:
3621:
3616:
3612:
3608:
3604:
3601:
3581:
3576:
3572:
3568:
3563:
3559:
3555:
3551:
3548:
3529:, and define
3523:
3510:
3505:
3502:
3495:
3492:
3490:
3486:
3481:
3479:
3475:
3470:
3453:
3448:
3443:
3440:
3437:
3432:
3427:
3424:
3421:
3416:
3413:
3406:
3403:
3400:
3395:
3390:
3385:
3380:
3377:
3374:
3369:
3366:
3359:
3356:
3353:
3348:
3343:
3340:
3337:
3332:
3327:
3322:
3319:
3307:
3303:
3300:
3293:
3286:
3280:
3276:
3272:
3268:
3263:
3253:
3246:
3239:
3232:
3225:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3173:
3169:
3165:
3162:
3157:
3153:
3149:
3146:
3144:
3137:
3127:
3124:
3113:
3105:
3101:
3097:
3092:
3088:
3084:
3079:
3075:
3068:
3063:
3059:
3055:
3052:
3050:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3013:
3009:
3005:
3002:
2997:
2993:
2989:
2987:
2980:
2970:
2967:
2952:
2951:
2950:
2945:
2938:
2903:
2894:
2891:
2885:
2883:
2872:
2869:
2859:
2850:
2847:
2841:
2839:
2828:
2825:
2811:
2810:
2809:
2804:
2781:
2772:
2769:
2763:
2754:
2751:
2745:
2737:
2728:
2725:
2719:
2716:
2707:
2704:
2689:
2688:
2687:
2685:
2681:
2677:
2666:
2659:
2649:
2641:
2639:
2637:
2633:
2628:
2626:
2622:
2603:
2594:
2591:
2585:
2576:
2573:
2563:
2562:
2561:
2514:
2509:
2501:
2498:
2492:
2487:
2477:
2474:
2467:
2462:
2452:
2449:
2442:
2437:
2427:
2424:
2417:
2412:
2408:
2404:
2399:
2394:
2390:
2386:
2381:
2376:
2372:
2368:
2363:
2358:
2354:
2345:
2342:
2339:
2336:
2327:
2324:
2318:
2309:
2306:
2300:
2291:
2288:
2282:
2279:
2276:
2271:
2267:
2263:
2258:
2254:
2250:
2245:
2241:
2228:
2227:
2226:
2220:
2201:
2178:
2173:
2169:
2165:
2162:
2157:
2147:
2144:
2137:
2132:
2122:
2119:
2112:
2107:
2097:
2094:
2087:
2082:
2078:
2074:
2069:
2065:
2061:
2056:
2052:
2044:
2043:
2042:
2036:
2028:
2023:
2019:
2016:
2012:
2008:
2004:
2001:
1997:
1993:
1992:
1977:
1974:
1964:
1961:
1951:
1948:
1941:
1931:
1928:
1918:
1915:
1908:
1898:
1895:
1885:
1882:
1871:
1870:
1866:
1865:
1850:
1847:
1837:
1834:
1824:
1821:
1814:
1804:
1801:
1791:
1788:
1781:
1771:
1768:
1758:
1755:
1744:
1743:
1739:
1735:
1731:
1730:
1726:
1718:
1704:
1695:
1692:
1686:
1677:
1674:
1668:
1659:
1656:
1650:
1641:
1638:
1632:
1623:
1620:
1614:
1605:
1602:
1592:
1591:
1587:
1586:
1585:
1571:
1569:
1567:
1563:
1544:
1534:
1531:
1493:
1483:
1480:
1473:
1470:
1466:
1456:
1453:
1415:
1405:
1402:
1391:
1390:
1389:
1387:
1378:
1351:
1347:
1343:
1338:
1334:
1330:
1325:
1321:
1317:
1312:
1308:
1300:
1299:
1298:
1295:
1289:
1271:
1263:
1262:complex plane
1258:
1256:
1252:
1248:
1242:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1187:
1174:
1170:
1166:
1161:
1157:
1153:
1148:
1144:
1140:
1129:
1125:
1121:
1116:
1112:
1108:
1103:
1099:
1092:
1084:
1080:
1076:
1071:
1067:
1063:
1058:
1054:
1050:
1039:
1035:
1031:
1026:
1022:
1018:
1013:
1009:
1002:
994:
990:
986:
981:
977:
973:
968:
964:
952:
948:
944:
939:
935:
931:
926:
922:
916:
903:
902:
897:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
779:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
728:
727:
710:
701:
698:
692:
688:
685:
682:
678:
669:
666:
660:
656:
653:
650:
646:
637:
634:
628:
624:
621:
608:
606:
598:
589:
584:
576:
574:
572:
571:extended side
568:
559:
552:
548:
547:Special Cases
544:
526:
523:
518:
513:
509:
501:
500:
499:
479:
475:
450:
447:
441:
437:
434:
428:
424:
420:
417:
415:
406:
403:
400:
391:
385:
381:
376:
369:
365:
358:
353:
347:
343:
338:
331:
327:
320:
318:
311:
307:
302:
298:
295:
292:
288:
281:
277:
267:
263:
258:
250:
246:
241:
239:
230:
222:
219:
216:
198:
194:
190:
182:
178:
174:
166:
162:
158:
154:
148:
143:
139:
137:
128:
126:
122:
118:
114:
109:
107:
103:
98:
96:
92:
87:
83:
82:
81:extended base
77:
76:
71:
67:
66:perpendicular
63:
62:
57:
53:
49:
45:
41:
33:
28:
22:
6654:
6622:
6604:
6593:. Retrieved
6590:Cut the Knot
6589:
6564:
6555:
6524:
6520:
6504:
6492:
6481:
6477:
6468:
6457:. Retrieved
6454:Cut The Knot
6453:
6443:. Quoted by
6427:
6421:
6405:
6396:
6392:
6372:
6368:
6355:
6345:
6319:
6314:
6306:
6239:
6231:
6226:
6218:
6210:
6202:
6197:
6190:Johnson 2007
6185:
6178:Johnson 2007
6173:
6162:
6157:
6139:
6133:
6125:
6120:
6113:Johnson 2007
6108:
6096:
6089:Johnson 2007
6084:
6072:
6065:Johnson 2007
6060:
6040:
6019:
6000:
5995:
5983:
5971:
5959:
5947:
5938:
5932:
5920:
5909:
5897:
5885:
5859:Johnson 2007
5854:
5843:. Retrieved
5839:the original
5819:Johnson 2007
5814:
5797:
5776:
5765:. Retrieved
5761:the original
5750:
5723:
5678:
5662:
5659:Isaac Newton
5657:(1619), and
5654:
5646:
5644:
5628:
5616:
5602:
5596:
5451:
5444:
5220:
5192:
4971:
4840:
4826:
4555:
4552:Area theorem
4437:
4430:
4423:
4410:
4403:
4396:
4393:
4324:circumradius
4306:
4176:
4142:
4075:
3902:
3885:
3728:
3710:
3703:
3521:
3506:
3500:
3496:
3493:
3482:
3471:
3304:
3298:
3291:
3284:
3278:
3274:
3261:
3257:
3251:
3244:
3237:
2944:circumradius
2922:
2800:
2665:circumcenter
2651:
2629:
2618:
2533:
2193:
2035:circumradius
2032:
2000:circumcenter
1740:through it.)
1725:polar circle
1575:
1561:
1559:
1386:free vectors
1384:by means of
1376:
1368:
1293:
1290:of triangle
1288:circumcenter
1259:
1243:
904:
898:
729:
609:
596:
594:
567:obtuse angle
564:
471:
196:
192:
188:
180:
176:
172:
164:
160:
156:
152:
135:
129:
110:
99:
90:
85:
79:
73:
59:
52:line segment
43:
37:
5619:, VII, 62;
2632:circumconic
2621:directrices
2033:Denote the
597:orthocenter
577:Orthocenter
6651:"Altitude"
6595:2022-12-17
6545:References
6459:2019-11-17
5890:Smart 1998
5845:2012-05-04
5809:, page 142
5767:2012-04-19
5728:Smart 1998
5609:Archimedes
5197:within an
3724:Euler line
3716:homothetic
3525:at vertex
2680:Euler line
2544:, so that
1572:Properties
581:See also:
478:hypotenuse
6656:MathWorld
6607:, Dover,
6554:(2007) ,
6527:: 60â96.
6501:"ZusÀtze"
6288:−
6272:−
6256:−
5655:Geometrie
5625:al-Nasawi
5555:⋅
5519:⋅
5485:⋅
5102:implies
4941:¯
4916:¯
4891:¯
4866:¯
4794:−
4781:−
4764:−
4751:−
4734:−
4721:−
4696:−
4645:−
4624:−
4603:−
4108:×
4100:×
4044:−
4029:−
4014:−
3860:−
3807:−
3754:−
3675:∩
3622:∩
3569:∩
3441:
3425:
3404:
3378:
3357:
3341:
3234:Triangle
3200:
3191:
3182:
3163:−
3132:¯
3069:−
3040:
3031:
3022:
3003:−
2975:¯
2899:¯
2877:¯
2855:¯
2833:¯
2777:¯
2759:¯
2733:¯
2712:¯
2625:parabolas
2599:¯
2581:¯
2482:¯
2457:¯
2432:¯
2332:¯
2314:¯
2296:¯
2219:excircles
2152:¯
2127:¯
2102:¯
1969:¯
1956:¯
1936:¯
1923:¯
1903:¯
1890:¯
1842:¯
1829:¯
1809:¯
1796:¯
1776:¯
1763:¯
1700:¯
1687:⋅
1682:¯
1664:¯
1651:⋅
1646:¯
1628:¯
1615:⋅
1610:¯
1539:→
1498:∑
1488:→
1474:⋅
1461:→
1420:∑
1410:→
1220:
1208:
1196:
1141:−
1109:−
1051:−
1032:−
974:−
945:−
875:
866:
860:−
854:
842:
833:
827:−
821:
809:
800:
794:−
788:
769:
757:
745:
706:¯
674:¯
642:¯
435:∴
392:⏞
354:⏞
117:congruent
6691:Category
6515:(1883).
6476:(1804).
6448:(2010).
6413:(1850).
6338:(1971).
6079:, p. 165
5978:, p. 123
5927:, p. 102
5892:, p. 182
5745:, p. 118
5730:, p. 156
5699:See also
4669:we have
4198:incircle
3831:″
3788:″
3745:″
3658:″
3605:″
3552:″
2937:inradius
2803:incenter
2684:midpoint
2658:centroid
2200:incircle
541: (
199: )
183: )
167: )
121:midpoint
58:(called
48:triangle
44:altitude
40:geometry
6558:, Dover
6399:: 1â36.
5633:al-Quhi
5593:History
5305:
5272:
5268:
5235:
4844:, then
4558:a, b, c
4200:radius
4179:a, b, c
3905:a, b, c
3483:In any
3267:oblique
2674:of the
2623:of all
2041:. Then
2013:of the
1738:cevians
1582:A, B, C
1578:D, E, F
1270:numbers
1266:A, B, C
1260:In the
612:A, B, C
134:(as in
6629:
6611:
6571:
6146:
6047:
5805:
5788:
5672:Later
5670:1680).
5613:Pappus
4112:height
3704:A"B"C"
2663:, the
2656:, the
2221:, and
545:; see
136:height
111:In an
64:) and
56:vertex
6505:Werke
6480:[
6426:. 3.
6418:(PDF)
6365:(PDF)
5716:Notes
4147:with
472:In a
50:is a
46:of a
42:, an
6627:ISBN
6609:ISBN
6569:ISBN
6510:See
6144:ISBN
6045:ISBN
5803:ISBN
5786:ISBN
5270:and
5225:and
5130:<
5081:<
4318:and
4162:and
4104:base
3907:and
3888:here
3472:The
2942:and
2886:>
2842:<
2619:The
2005:The
1994:The
1576:Let
899:and
610:Let
487:and
185:and
86:foot
75:base
70:edge
61:apex
6529:doi
6432:doi
6377:doi
6305:,"
5642:).
5637:fl.
4841:ABC
4827:If
4570:, h
4566:, h
4394:If
4191:, h
4187:, h
4151:or
4140:).
3711:ABC
3535:, L
3522:ABC
3501:ABC
3438:sec
3422:sec
3401:sec
3375:sec
3354:sec
3338:sec
3299:ABC
3292:DEF
3285:DEF
3277:or
3265:is
3262:ABC
3252:ABC
3245:DEF
3238:abc
3197:cos
3188:cos
3179:cos
3037:cos
3028:cos
3019:cos
2212:, r
2208:, r
1377:ABC
1294:ABC
1281:, z
1277:, z
1217:tan
1205:tan
1193:tan
872:sin
863:sin
851:cos
839:sin
830:sin
818:cos
806:sin
797:sin
785:cos
766:sec
754:sec
742:sec
38:In
6693::
6653:.
6588:.
6584:.
6523:.
6519:.
6503:.
6452:.
6428:37
6420:.
6395:.
6391:.
6373:60
6371:.
6367:.
6327:^
6039:.
6027:^
6011:^
5876:,
5866:^
5826:^
5735:^
5668:c.
5621:c.
5449:.
5205:.
4834:AD
4536:1.
4436:,
4429:,
4409:,
4402:,
3890:.
3883:.
3726:.
3480:.
3302:.
2949:,
2935:,
2931:,
2927:,
2638:.
2630:A
2560::
2557:HP
2547:AD
2537:AD
2202:,
2166:12
1978:2.
1851:1.
1568:.
1257:.
549:,
195:,
191:,
179:,
175:,
169:,
163:,
159:,
108:.
97:.
6683:.
6659:.
6598:.
6537:.
6531::
6525:1
6462:.
6438:.
6434::
6397:6
6383:.
6379::
6291:2
6284:d
6280:=
6275:2
6268:b
6264:+
6259:2
6252:a
6053:.
5880:.
5848:.
5770:.
5666:(
5635:(
5615:(
5569:B
5566:A
5561:C
5558:B
5552:C
5549:A
5542:=
5535:D
5532:C
5525:D
5522:C
5516:B
5513:A
5507:2
5504:1
5498:=
5491:C
5488:B
5482:C
5479:A
5473:2
5470:1
5430:.
5423:2
5419:b
5415:1
5410:+
5403:2
5399:a
5395:1
5390:=
5383:2
5378:b
5374:h
5370:1
5365:+
5358:2
5353:a
5349:h
5345:1
5340:=
5333:2
5328:c
5324:h
5320:1
5293:a
5290:=
5285:b
5281:h
5256:b
5253:=
5248:a
5244:h
5231:c
5227:b
5223:a
5195:P
5179:.
5163:c
5159:h
5155:1
5150:+
5143:b
5139:h
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5119:h
5115:1
5090:c
5087:+
5084:b
5078:a
5056:c
5052:h
5048:c
5042:2
5039:1
5033:=
5028:b
5024:h
5020:b
5014:2
5011:1
5005:=
5000:a
4996:h
4992:a
4986:2
4983:1
4952:.
4947:2
4937:E
4934:C
4927:+
4922:2
4912:B
4909:A
4902:=
4897:2
4887:B
4884:E
4877:+
4872:2
4862:C
4859:A
4839:âł
4829:E
4807:.
4802:)
4797:1
4789:c
4785:h
4778:H
4775:(
4772:)
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4759:b
4755:h
4748:H
4745:(
4742:)
4737:1
4729:a
4725:h
4718:H
4715:(
4712:H
4707:4
4704:=
4699:1
4691:a
4688:e
4685:r
4682:A
4654:2
4648:1
4640:c
4636:h
4632:+
4627:1
4619:b
4615:h
4611:+
4606:1
4598:a
4594:h
4586:=
4583:H
4572:c
4568:b
4564:a
4562:h
4533:=
4526:3
4522:h
4516:3
4512:p
4506:+
4499:2
4495:h
4489:2
4485:p
4479:+
4472:1
4468:h
4462:1
4458:p
4441:3
4438:h
4434:2
4431:h
4427:1
4424:h
4419:P
4414:3
4411:p
4407:2
4404:p
4400:1
4397:p
4374:.
4368:R
4365:2
4360:c
4357:b
4351:=
4346:a
4342:h
4328:R
4320:c
4316:b
4311:a
4309:h
4286:.
4279:c
4275:h
4271:1
4266:+
4259:b
4255:h
4251:1
4246:+
4239:a
4235:h
4231:1
4226:=
4221:r
4218:1
4202:r
4193:c
4189:b
4185:a
4183:h
4166:c
4164:h
4159:b
4157:h
4153:c
4149:b
4145:a
4138:a
4134:A
4130:a
4116:,
4094:2
4091:1
4061:.
4056:a
4050:)
4047:c
4041:s
4038:(
4035:)
4032:b
4026:s
4023:(
4020:)
4017:a
4011:s
4008:(
4005:s
4000:2
3994:=
3989:a
3985:h
3971:a
3957:,
3954:)
3951:c
3948:+
3945:b
3942:+
3939:a
3936:(
3930:2
3927:1
3921:=
3918:s
3863:c
3855::
3850:b
3845::
3840:a
3835:=
3828:C
3820:c
3815::
3810:b
3802::
3797:a
3792:=
3785:B
3777:c
3772::
3767:b
3762::
3757:a
3749:=
3742:A
3709:âł
3702:âł
3688:.
3683:A
3679:L
3670:C
3666:L
3662:=
3655:C
3635:,
3630:A
3626:L
3617:C
3613:L
3609:=
3602:B
3582:,
3577:C
3573:L
3564:B
3560:L
3556:=
3549:A
3537:C
3533:B
3531:L
3527:A
3520:âł
3515:A
3513:L
3499:âł
3454:0
3449::
3444:B
3433::
3428:A
3417:=
3414:F
3407:C
3396::
3391:0
3386::
3381:A
3370:=
3367:E
3360:C
3349::
3344:B
3333::
3328:0
3323:=
3320:D
3297:âł
3290:âł
3283:âł
3260:âł
3250:âł
3243:âł
3236:âł
3206:.
3203:C
3194:B
3185:A
3174:2
3170:R
3166:4
3158:2
3154:r
3150:2
3147:=
3138:2
3128:I
3125:H
3114:,
3111:)
3106:2
3102:c
3098:+
3093:2
3089:b
3085:+
3080:2
3076:a
3072:(
3064:2
3060:R
3056:9
3053:=
3043:C
3034:B
3025:A
3014:2
3010:R
3006:8
2998:2
2994:R
2990:=
2981:2
2971:H
2968:O
2947:R
2940:r
2933:c
2929:b
2925:a
2904:.
2895:G
2892:I
2873:G
2870:H
2860:,
2851:G
2848:H
2829:I
2826:H
2806:I
2782:.
2773:H
2770:G
2764:=
2755:G
2752:O
2746:2
2738:,
2729:H
2726:N
2720:2
2717:=
2708:H
2705:O
2672:N
2668:O
2661:G
2654:H
2604:.
2595:P
2592:D
2586:=
2577:D
2574:H
2552:D
2542:P
2515:.
2510:2
2506:)
2502:R
2499:2
2496:(
2493:+
2488:2
2478:H
2475:C
2468:+
2463:2
2453:H
2450:B
2443:+
2438:2
2428:H
2425:A
2418:=
2413:2
2409:r
2405:+
2400:2
2395:c
2391:r
2387:+
2382:2
2377:b
2373:r
2369:+
2364:2
2359:a
2355:r
2346:,
2343:R
2340:2
2337:+
2328:H
2325:C
2319:+
2310:H
2307:B
2301:+
2292:H
2289:A
2283:=
2280:r
2277:+
2272:c
2268:r
2264:+
2259:b
2255:r
2251:+
2246:a
2242:r
2223:R
2214:c
2210:b
2206:a
2204:r
2196:r
2179:.
2174:2
2170:R
2163:=
2158:2
2148:H
2145:C
2138:+
2133:2
2123:H
2120:B
2113:+
2108:2
2098:H
2095:A
2088:+
2083:2
2079:c
2075:+
2070:2
2066:b
2062:+
2057:2
2053:a
2039:R
2017:.
1975:=
1965:F
1962:C
1952:H
1949:C
1942:+
1932:E
1929:B
1919:H
1916:B
1909:+
1899:D
1896:A
1886:H
1883:A
1848:=
1838:F
1835:C
1825:F
1822:H
1815:+
1805:E
1802:B
1792:E
1789:H
1782:+
1772:D
1769:A
1759:D
1756:H
1727:.
1721:H
1705:.
1696:F
1693:H
1678:H
1675:C
1669:=
1660:E
1657:H
1642:H
1639:B
1633:=
1624:D
1621:H
1606:H
1603:A
1545:.
1535:A
1532:H
1520:c
1517:i
1514:l
1511:c
1508:y
1505:c
1494:=
1484:O
1481:H
1471:2
1467:,
1457:A
1454:O
1442:c
1439:i
1436:l
1433:c
1430:y
1427:c
1416:=
1406:H
1403:O
1382:H
1375:âł
1371:H
1352:C
1348:z
1344:+
1339:B
1335:z
1331:+
1326:A
1322:z
1318:=
1313:H
1309:z
1292:âł
1283:C
1279:B
1275:A
1273:z
1226:.
1223:C
1214::
1211:B
1202::
1199:A
1190:=
1180:)
1175:2
1171:c
1167:+
1162:2
1158:b
1154:+
1149:2
1145:a
1138:(
1135:)
1130:2
1126:c
1122:+
1117:2
1113:b
1104:2
1100:a
1096:(
1093::
1090:)
1085:2
1081:c
1077:+
1072:2
1068:b
1064:+
1059:2
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1048:(
1045:)
1040:2
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1027:2
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1019:+
1014:2
1010:a
1006:(
1003::
1000:)
995:2
991:c
987:+
982:2
978:b
969:2
965:a
961:(
958:)
953:2
949:c
940:2
936:b
932:+
927:2
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919:(
881:,
878:B
869:A
857:C
848::
845:A
836:C
824:B
815::
812:C
803:B
791:A
782:=
772:C
763::
760:B
751::
748:A
711:|
702:B
699:A
693:|
689:=
686:c
683:,
679:|
670:A
667:C
661:|
657:=
654:b
651:,
647:|
638:C
635:B
629:|
625:=
622:a
601:H
553:)
527:q
524:p
519:=
514:c
510:h
495:c
493:h
489:q
485:p
481:c
451:q
448:p
442:=
438:h
429:2
425:h
421:2
418:=
407:q
404:p
401:2
386:2
382:q
377:+
370:2
366:h
359:+
348:2
344:h
339:+
332:2
328:p
321:=
312:2
308:q
303:+
299:q
296:p
293:2
289:+
282:2
278:p
268:2
264:s
259:+
251:2
247:r
242:=
231:2
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223:q
220:+
217:p
214:(
201:,
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187:(
181:h
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