20:
1142:
908:
919:
642:
3899:; and there is a circumsphere passing through all of the vertices, whose center is the circumcenter. These points define the "Euler line" of a tetrahedron analogous to that of a triangle. The centroid is the midpoint between its Monge point and circumcenter along this line. The center of the
3629:
1137:{\displaystyle 3\cdot {\vec {GO}}=\left(\sum \limits _{\scriptstyle {\rm {cyc}}}{\vec {GA}}\right)+\left(\sum \limits _{\scriptstyle {\rm {cyc}}}{\vec {AO}}\right)=0-\left(\sum \limits _{\scriptstyle {\rm {cyc}}}{\vec {OA}}\right)=-{\vec {OH}}.}
903:{\displaystyle {\vec {GO}}={\vec {GA}}+{\vec {AO}}\,{\mbox{(in triangle }}AGO{\mbox{)}},\,{\vec {GO}}={\vec {GB}}+{\vec {BO}}\,{\mbox{(in triangle }}BGO{\mbox{)}},\,{\vec {GO}}={\vec {GC}}+{\vec {CO}}\,{\mbox{(in triangle }}CGO{\mbox{)}}.}
2211:
3357:
184:, although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them.
2897:
2517:
2392:
3045:
2740:
2051:
1852:
1643:
631:
527:
1484:
447:
1743:
1205:
3715:—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its
3433:
2089:
2592:
2273:
1278:
are put together into a single proof. However, most of the proofs of the problem of
Sylvester rely on the fundamental properties of free vectors, independently of the Euler line.
3677:
3446:
4362:
It is well known that the incenter of a
Euclidean triangle lies on its Euler line connecting the centroid and the circumcenter if and only if the triangle is isosceles
1381:
1342:
3175:
3145:
3116:
3202:
3074:
2923:
2766:
2618:
4788:
4156:
3919:(plural of simplex). For example, every polygon is a simplicial polytope. The Euler line associated to such a polytope is the line determined by its centroid and
267:
4199:
4179:
4130:
4110:
4087:
4067:
4047:
4027:
4007:
3984:
3964:
3944:
1265:
1245:
1225:
474:
364:
336:
313:
290:
2097:
3210:
2777:
2400:
2278:
2934:
2629:
1901:
4734:
2057:
222:
at the reference triangle's vertices. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle. The
4596:
4449:
Dörrie, Heinrich, "100 Great
Problems of Elementary Mathematics. Their History and Solution". Dover Publications, Inc., New York, 1965,
1751:
4310:
1545:
539:
4454:
479:
1275:
530:
4673:
1407:
372:
4482:
Wladimir G. Boskoff, Laurent¸iu
Homentcovschi, and Bogdan D. Suceava, "Gossard's Perspector and Projective Consequences",
1520:
Furthermore, the squared distance between the centroid and the circumcenter along the Euler line is less than the squared
453:
1654:
3772:
3085:
1150:
4214:
3759:
are in the same proportions, though in the opposite order, as the sides) is perpendicular to one of the medians.
3363:
3895:. Seven lines associated with a tetrahedron are concurrent at its centroid; its six midplanes intersect at its
211:, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
3888:
2062:
128:
of the triangle, and it passes through several important points determined from the triangle, including the
125:
2544:
4328:
2228:
1391:
3624:{\displaystyle m_{E}=-{\frac {m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}+3}{m_{1}+m_{2}+m_{3}+3m_{1}m_{2}m_{3}}}.}
1892:
1401:
of the nine-point circle lies along the Euler line midway between the orthocenter and the circumcenter:
4594:
Tabachnikov, Serge; Tsukerman, Emmanuel (May 2014), "Circumcenter of Mass and
Generalized Euler Line",
3690:
inscribed in a given triangle is formed by two lines perpendicular to the given triangle's Euler line.
4686:
4326:
Edmonds, Allan L.; Hajja, Mowaffaq; Martini, Horst (2008), "Orthocentric simplices and biregularity",
4380:
3920:
3687:
3641:
270:
223:
121:
4739:
4695:
4669:
4499:
4279:, ser. I, vol. XXVI, pp. 139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953,
3912:
3900:
3814:
3752:
3716:
3440:
Thus the slope of the Euler line (if finite) is expressible in terms of the slopes of the sides as
2928:
316:
215:
188:
55:
4719:
4730:
4623:
4605:
4533:
4405:
4397:
4353:
3732:
2523:
208:
200:
113:
4209:
A triangle's
Kiepert parabola is the unique parabola that is tangent to the sides (two of them
4761:
4450:
4306:
4262:
3851:
3822:
3756:
3708:
2771:
181:
151:
The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the
145:
41:
32:
28:
2394:
every point on the Euler line, except the orthocenter, is given by the trilinear coordinates
1348:
1309:
4715:
4678:
4615:
4555:
4525:
4389:
4337:
4300:
3803:
3736:
3150:
3120:
3091:
343:
92:
4349:
4283:
3180:
3050:
4682:
4497:
Francisco Javier Garc ́ıa Capita ́n, "Locus of
Centroids of Similar Inscribed Triangles",
4345:
4280:
3892:
2902:
2745:
2597:
227:
192:
177:
173:
172:
Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are
4690:
4135:
246:
4703:
4640:
4504:
4487:
4258:
4184:
4164:
4115:
4095:
4072:
4052:
4032:
4012:
3992:
3969:
3949:
3929:
3704:
1250:
1230:
1210:
459:
349:
321:
298:
275:
86:
4699:
4641:
Scimemi, Benedetto, "Simple
Relations Regarding the Steiner Inellipse of a Triangle",
4568:
2206:{\displaystyle (\tan C-\tan B)\alpha +(\tan A-\tan C)\beta +(\tan B-\tan A)\gamma =0.}
4782:
4409:
4357:
4210:
3839:
152:
4627:
4708:
4466:
Scott, J.A., "Some examples of the use of areal coordinates in triangle geometry",
4287:
3719:, falls on the right-angled vertex while its circumcenter, the intersection of its
2538:
1521:
219:
196:
141:
133:
69:
4660:
An interactive applet showing several triangle centers that lies on the Euler line
19:
3352:{\displaystyle m_{1}m_{2}+m_{1}m_{3}+m_{1}m_{E}+m_{2}m_{3}+m_{2}m_{E}+m_{3}m_{E}}
1489:
Thus the Euler line could be repositioned on a number line with the circumcenter
4748:
3896:
3884:
3878:
3818:
156:
129:
59:
4764:
4665:
3821:
of the triangle with vertices at the other three points) are concurrent at the
2892:{\displaystyle \cos A+2\cos B\cos C:\cos B+2\cos C\cos A:\cos C+2\cos A\cos B,}
4743:
4619:
4393:
4378:
Leversha, Gerry; Smith, G. C. (November 2007), "Euler and triangle geometry",
4341:
3712:
2512:{\displaystyle \cos A+t\cos B\cos C:\cos B+t\cos C\cos A:\cos C+t\cos A\cos B}
2387:{\displaystyle \sec A:\sec B:\sec C=\cos B\cos C:\cos C\cos A:\cos A\cos B),}
4769:
4659:
3858:
3720:
3040:{\displaystyle \cos A-\cos B\cos C:\cos B-\cos C\cos A:\cos C-\cos A\cos B,}
2735:{\displaystyle \cos A+\cos B\cos C:\cos B+\cos C\cos A:\cos C+\cos A\cos B,}
2046:{\displaystyle \sin(2A)\sin(B-C)x+\sin(2B)\sin(C-A)y+\sin(2C)\sin(A-B)z=0.}
1527:
by an amount equal to one-ninth the sum of the squares of the side lengths
207:
generally does not lie on the Euler line; it is on the Euler line only for
3740:
2623:
1303:
and is twice as far from the orthocenter as it is from the circumcenter:
293:
204:
137:
117:
78:
45:
4401:
4302:
Geometry Turned On: Dynamic
Software in Learning, Teaching, and Research
4753:
4537:
3916:
4263:"Solutio facilis problematum quorundam geometricorum difficillimorum"
4529:
4239:
Kimberling, Clark (1998). "Triangle centers and central triangles".
68: Perpendicular lines from the side midpoints (intersect at the
4516:
Parry, C. F. (1991), "Steiner–Lehmus and the automedian triangle",
2221:
Another way to represent the Euler line is in terms of a parameter
4610:
4457:, pages 141 (Euler's Straight Line) and 142 (Problem of Sylvester)
18:
3634:
Moreover, the Euler line is parallel to an acute triangle's side
1847:{\displaystyle GH^{2}=4R^{2}-{\tfrac {4}{9}}(a^{2}+b^{2}+c^{2}).}
3788:. The Euler lines of the 10 triangles with vertices chosen from
1638:{\displaystyle GO^{2}=R^{2}-{\tfrac {1}{9}}(a^{2}+b^{2}+c^{2}).}
626:{\displaystyle {\vec {OH}}={\vec {OA}}+{\vec {OB}}+{\vec {OC}}.}
4267:
Novi
Commentarii Academiae Scientarum Imperialis Petropolitanae
3923:. This definition of an Euler line generalizes the ones above.
4265:[Easy solution of some difficult geometric problems].
3879:
Tetrahedron § Properties analogous to those of a triangle
2225:. Starting with the circumcenter (with trilinear coordinates
913:
By adding these three relations, term by term, we obtain that
104:
98:
1879:
denote the vertex angles of the reference triangle, and let
522:{\displaystyle {\frac {1}{3}}:{\frac {1}{3}}:{\frac {1}{3}}}
187:
Other notable points that lie on the Euler line include the
4437:, Dover Publications, 2007 (orig. Barnes & Noble 1952).
4556:
http://forumgeom.fau.edu/FG2009volume9/FG200924index.html
4305:. The Mathematical Association of America. pp. 3–4.
4550:
Beluhov, Nikolai Ivanov. "Ten concurrent Euler lines",
1788:
1579:
1479:{\displaystyle ON=NH,\quad OG=2\cdot GN,\quad NH=3GN.}
1069:
1011:
959:
891:
875:
806:
790:
721:
705:
442:{\displaystyle {\vec {GA}}+{\vec {GB}}+{\vec {GC}}=0.}
4687:
The quasi-Euler line of a quadrilateral and a hexagon
4187:
4167:
4138:
4118:
4098:
4075:
4055:
4035:
4015:
3995:
3972:
3952:
3932:
3644:
3449:
3366:
3213:
3183:
3153:
3123:
3094:
3053:
2937:
2905:
2780:
2748:
2632:
2600:
2547:
2403:
2281:
2231:
2100:
2065:
1904:
1754:
1657:
1548:
1410:
1351:
1312:
1253:
1233:
1213:
1153:
922:
645:
542:
482:
462:
375:
352:
324:
301:
278:
249:
4569:"On Two Remarkable Lines Related to a Quadrilateral"
4505:
http://forumgeom.fau.edu/FG2016volume16/FG201631.pdf
230:
and tangential triangles is also on the Euler line.
101:
3813:The Euler lines of the four triangles formed by an
3723:of sides, falls on the midpoint of the hypotenuse.
218:of a reference triangle is tangent to the latter's
95:
4193:
4173:
4150:
4124:
4104:
4081:
4061:
4041:
4021:
4001:
3978:
3958:
3938:
3671:
3623:
3427:
3351:
3196:
3169:
3139:
3110:
3088:, denote the slopes of the sides of a triangle as
3068:
3039:
2917:
2891:
2760:
2734:
2612:
2586:
2511:
2386:
2267:
2205:
2083:
2045:
1846:
1738:{\displaystyle OH^{2}=9R^{2}-(a^{2}+b^{2}+c^{2});}
1737:
1637:
1478:
1375:
1336:
1259:
1239:
1219:
1199:
1136:
902:
625:
521:
468:
441:
358:
330:
307:
284:
261:
4213:) of the triangle and has the Euler line as its
3763:Systems of triangles with concurrent Euler lines
2526:of the trilinears of these two points, for some
346:. We start by stating the prerequisites. First,
636:Now, using the vector addition, we deduce that
4679:Nine-point conic and Euler line generalization
3204:. Then these slopes are related according to
1200:{\displaystyle 3\cdot {\vec {OG}}={\vec {OH}}}
8:
4299:Schattschneider, Doris; King, James (1997).
3817:(a set of four points such that each is the
269:be a triangle. A proof of the fact that the
3682:Relation to inscribed equilateral triangles
3428:{\displaystyle +3m_{1}m_{2}m_{3}m_{E}+3=0.}
3177:and denote the slope of its Euler line as
4609:
4186:
4166:
4137:
4117:
4097:
4074:
4054:
4034:
4014:
3994:
3971:
3951:
3931:
3643:
3609:
3599:
3589:
3573:
3560:
3547:
3529:
3519:
3506:
3496:
3483:
3473:
3466:
3454:
3448:
3407:
3397:
3387:
3377:
3365:
3343:
3333:
3320:
3310:
3297:
3287:
3274:
3264:
3251:
3241:
3228:
3218:
3212:
3188:
3182:
3158:
3152:
3128:
3122:
3099:
3093:
3052:
2936:
2904:
2779:
2747:
2631:
2599:
2546:
2402:
2280:
2230:
2099:
2064:
1903:
1895:; then an equation for the Euler line is
1832:
1819:
1806:
1787:
1778:
1762:
1753:
1723:
1710:
1697:
1681:
1665:
1656:
1623:
1610:
1597:
1578:
1569:
1556:
1547:
1409:
1350:
1311:
1252:
1232:
1212:
1181:
1180:
1161:
1160:
1152:
1115:
1114:
1087:
1086:
1071:
1070:
1068:
1029:
1028:
1013:
1012:
1010:
977:
976:
961:
960:
958:
930:
929:
921:
890:
874:
873:
857:
856:
837:
836:
817:
816:
815:
805:
789:
788:
772:
771:
752:
751:
732:
731:
730:
720:
704:
703:
687:
686:
667:
666:
647:
646:
644:
604:
603:
584:
583:
564:
563:
544:
543:
541:
509:
496:
483:
481:
461:
417:
416:
397:
396:
377:
376:
374:
351:
323:
300:
277:
248:
176:. This property is also true for another
4445:
4443:
4226:
2275:) and the orthocenter (with trilinears
2084:{\displaystyle \alpha :\beta :\gamma }
4789:Straight lines defined for a triangle
4478:
4476:
3861:in this order on the Euler line, and
3047:corresponding to the parameter value
2899:corresponding to the parameter value
2742:corresponding to the parameter value
2594:corresponding to the parameter value
2587:{\displaystyle \cos A:\cos B:\cos C,}
7:
4720:"Triangle centers on the Euler line"
4429:
4427:
4425:
4423:
4421:
4419:
4373:
4371:
4112:has a center of rotational symmetry
3707:, the Euler line coincides with the
2268:{\displaystyle \cos A:\cos B:\cos C}
452:This follows from the fact that the
4683:A further Euler line generalization
4597:Discrete and Computational Geometry
4234:
4232:
4230:
3915:is a polytope whose facets are all
1065:
1007:
955:
4670:"Non-Euclidean Triangle Continuum"
4161:3. If all but one of the sides of
4009:has a line of reflection symmetry
3966:is sensitive to the symmetries of
3891:object bounded by four triangular
2056:An equation for the Euler line in
1078:
1075:
1072:
1020:
1017:
1014:
968:
965:
962:
163:Triangle centers on the Euler line
14:
4744:"Triangles have a Magic Highway"
4201:is orthogonal to the last side.
3825:common to all of the triangles.
1493:at the location 0, the centroid
454:absolute barycentric coordinates
91:
16:Line constructed from a triangle
3739:. In an isosceles triangle the
3672:{\displaystyle \tan B\tan C=3.}
1454:
1429:
1291:On the Euler line the centroid
1267:(in this order) are collinear.
4674:Wolfram Demonstrations Project
3686:The locus of the centroids of
2378:
2191:
2167:
2158:
2134:
2125:
2101:
2031:
2019:
2010:
2001:
1986:
1974:
1965:
1956:
1941:
1929:
1920:
1911:
1838:
1799:
1729:
1690:
1629:
1590:
1191:
1171:
1125:
1097:
1039:
987:
940:
867:
847:
827:
782:
762:
742:
697:
677:
657:
614:
594:
574:
554:
427:
407:
387:
1:
4704:Euler Line and 9-Point Circle
4486:, Volume 13 (2013), 169–184.
3946:is a polygon. The Euler line
3903:also lies on the Euler line.
4700:Altitudes and the Euler Line
3806:at the centroid of triangle
1501:, the nine-point center at 3
1295:is between the circumcenter
4470:83, November 1999, 472-477.
3086:Cartesian coordinate system
4805:
4567:Myakishev, Alexei (2006),
4433:Altshiller-Court, Nathan,
3876:
1207:, and so the three points
4691:Dynamic Geometry Sketches
4620:10.1007/s00454-014-9597-2
4394:10.1017/S0025557200182087
4342:10.1007/s00025-008-0294-4
3743:falls on the Euler line.
2217:Parametric representation
1287:Distances between centers
27: Euler's line, with
4518:The Mathematical Gazette
4181:have equal length, then
3773:Fermat–Torricelli points
3986:in the following ways:
3842:, the quasiorthocenter
3721:perpendicular bisectors
2058:barycentric coordinates
1891:be a variable point in
1376:{\displaystyle OH=3GO.}
1337:{\displaystyle GH=2GO;}
366:satisfies the relation
4554:9, 2009, pp. 271–274.
4329:Results in Mathematics
4241:Congressus Numerantium
4195:
4175:
4152:
4126:
4106:
4083:
4063:
4043:
4023:
4003:
3980:
3960:
3940:
3846:, the "area centroid"
3673:
3625:
3429:
3353:
3198:
3171:
3170:{\displaystyle m_{3},}
3141:
3140:{\displaystyle m_{2},}
3112:
3111:{\displaystyle m_{1},}
3070:
3041:
2919:
2893:
2762:
2736:
2614:
2588:
2513:
2388:
2269:
2207:
2085:
2047:
1848:
1739:
1639:
1513:for some scale factor
1505:, and the orthocenter
1480:
1392:orthocentroidal circle
1377:
1338:
1270:In Dörrie's book, the
1261:
1241:
1221:
1201:
1138:
904:
627:
523:
470:
443:
360:
332:
309:
286:
263:
144:and the center of the
74:
4205:Related constructions
4196:
4176:
4153:
4127:
4107:
4084:
4064:
4044:
4024:
4004:
3981:
3961:
3941:
3751:The Euler line of an
3731:The Euler line of an
3688:equilateral triangles
3674:
3626:
3430:
3354:
3199:
3197:{\displaystyle m_{E}}
3172:
3142:
3113:
3071:
3069:{\displaystyle t=-1.}
3042:
2920:
2894:
2763:
2737:
2615:
2589:
2514:
2389:
2270:
2208:
2086:
2048:
1893:trilinear coordinates
1849:
1740:
1640:
1481:
1390:is a diameter of the
1378:
1339:
1262:
1242:
1222:
1202:
1139:
905:
628:
524:
471:
444:
361:
333:
310:
287:
264:
22:
4742:(February 1, 2016),
4740:Stankova, Zvezdelina
4696:Bogomolny, Alexander
4468:Mathematical Gazette
4381:Mathematical Gazette
4185:
4165:
4136:
4116:
4096:
4073:
4053:
4033:
4013:
3993:
3970:
3950:
3930:
3921:circumcenter of mass
3840:convex quadrilateral
3767:Consider a triangle
3694:In special triangles
3642:
3447:
3364:
3211:
3181:
3151:
3121:
3092:
3051:
2935:
2918:{\displaystyle t=2.}
2903:
2778:
2761:{\displaystyle t=1.}
2746:
2630:
2613:{\displaystyle t=0.}
2598:
2545:
2401:
2279:
2229:
2098:
2063:
1902:
1752:
1655:
1546:
1408:
1349:
1310:
1299:and the orthocenter
1276:problem of Sylvester
1251:
1231:
1211:
1151:
920:
643:
540:
531:problem of Sylvester
480:
460:
373:
350:
322:
299:
276:
247:
224:center of similitude
116:determined from any
4643:Forum Geometricorum
4576:Forum Geometricorum
4552:Forum Geometricorum
4503:16, 2016, 257–267 .
4500:Forum Geometricorum
4484:Forum Geometricorum
4286:. Summarized at:
4151:{\displaystyle E=C}
3913:simplicial polytope
3907:Simplicial polytope
3901:twelve-point sphere
3815:orthocentric system
3753:automedian triangle
3747:Automedian triangle
3735:coincides with the
2929:de Longchamps point
262:{\displaystyle ABC}
216:tangential triangle
209:isosceles triangles
189:de Longchamps point
4762:Weisstein, Eric W.
4288:Dartmouth College.
4191:
4171:
4148:
4122:
4102:
4079:
4059:
4039:
4019:
3999:
3976:
3956:
3936:
3733:isosceles triangle
3727:Isosceles triangle
3669:
3621:
3425:
3349:
3194:
3167:
3137:
3108:
3066:
3037:
2915:
2889:
2758:
2732:
2610:
2584:
2524:linear combination
2509:
2384:
2265:
2203:
2081:
2043:
1844:
1797:
1735:
1635:
1588:
1476:
1373:
1334:
1257:
1237:
1217:
1197:
1134:
1085:
1083:
1027:
1025:
975:
973:
900:
895:
879:
877:(in triangle
810:
794:
792:(in triangle
725:
709:
707:(in triangle
623:
519:
466:
439:
356:
328:
305:
282:
259:
201:Gossard perspector
168:Individual centers
75:
58:(intersect at the
44:(intersect at the
4716:Kimberling, Clark
4194:{\displaystyle E}
4174:{\displaystyle P}
4125:{\displaystyle C}
4105:{\displaystyle P}
4082:{\displaystyle L}
4062:{\displaystyle L}
4042:{\displaystyle E}
4022:{\displaystyle L}
4002:{\displaystyle P}
3979:{\displaystyle P}
3959:{\displaystyle E}
3939:{\displaystyle P}
3889:three-dimensional
3852:quasicircumcenter
3823:nine-point center
3616:
2772:nine-point center
1796:
1587:
1260:{\displaystyle H}
1240:{\displaystyle G}
1220:{\displaystyle O}
1194:
1174:
1128:
1100:
1064:
1042:
1006:
990:
954:
943:
894:
878:
870:
850:
830:
809:
793:
785:
765:
745:
724:
708:
700:
680:
660:
617:
597:
577:
557:
517:
504:
491:
469:{\displaystyle G}
430:
410:
390:
359:{\displaystyle G}
331:{\displaystyle H}
308:{\displaystyle G}
285:{\displaystyle O}
182:nine-point center
148:of the triangle.
146:nine-point circle
33:nine-point circle
4796:
4775:
4774:
4756:
4726:
4724:Triangle Centers
4647:
4645:10, 2010: 55–77.
4638:
4632:
4630:
4613:
4591:
4585:
4583:
4573:
4564:
4558:
4548:
4542:
4540:
4524:(472): 151–154,
4513:
4507:
4495:
4489:
4480:
4471:
4464:
4458:
4447:
4438:
4435:College Geometry
4431:
4414:
4412:
4388:(522): 436–452,
4375:
4366:
4364:
4323:
4317:
4316:
4296:
4290:
4274:
4273:: 103–123. E325.
4255:
4249:
4248:
4236:
4200:
4198:
4197:
4192:
4180:
4178:
4177:
4172:
4157:
4155:
4154:
4149:
4131:
4129:
4128:
4123:
4111:
4109:
4108:
4103:
4088:
4086:
4085:
4080:
4068:
4066:
4065:
4060:
4048:
4046:
4045:
4040:
4028:
4026:
4025:
4020:
4008:
4006:
4005:
4000:
3985:
3983:
3982:
3977:
3965:
3963:
3962:
3957:
3945:
3943:
3942:
3937:
3737:axis of symmetry
3678:
3676:
3675:
3670:
3630:
3628:
3627:
3622:
3617:
3615:
3614:
3613:
3604:
3603:
3594:
3593:
3578:
3577:
3565:
3564:
3552:
3551:
3541:
3534:
3533:
3524:
3523:
3511:
3510:
3501:
3500:
3488:
3487:
3478:
3477:
3467:
3459:
3458:
3434:
3432:
3431:
3426:
3412:
3411:
3402:
3401:
3392:
3391:
3382:
3381:
3358:
3356:
3355:
3350:
3348:
3347:
3338:
3337:
3325:
3324:
3315:
3314:
3302:
3301:
3292:
3291:
3279:
3278:
3269:
3268:
3256:
3255:
3246:
3245:
3233:
3232:
3223:
3222:
3203:
3201:
3200:
3195:
3193:
3192:
3176:
3174:
3173:
3168:
3163:
3162:
3146:
3144:
3143:
3138:
3133:
3132:
3117:
3115:
3114:
3109:
3104:
3103:
3075:
3073:
3072:
3067:
3046:
3044:
3043:
3038:
2924:
2922:
2921:
2916:
2898:
2896:
2895:
2890:
2767:
2765:
2764:
2759:
2741:
2739:
2738:
2733:
2619:
2617:
2616:
2611:
2593:
2591:
2590:
2585:
2518:
2516:
2515:
2510:
2393:
2391:
2390:
2385:
2274:
2272:
2271:
2266:
2212:
2210:
2209:
2204:
2090:
2088:
2087:
2082:
2052:
2050:
2049:
2044:
1853:
1851:
1850:
1845:
1837:
1836:
1824:
1823:
1811:
1810:
1798:
1789:
1783:
1782:
1767:
1766:
1744:
1742:
1741:
1736:
1728:
1727:
1715:
1714:
1702:
1701:
1686:
1685:
1670:
1669:
1644:
1642:
1641:
1636:
1628:
1627:
1615:
1614:
1602:
1601:
1589:
1580:
1574:
1573:
1561:
1560:
1485:
1483:
1482:
1477:
1382:
1380:
1379:
1374:
1343:
1341:
1340:
1335:
1266:
1264:
1263:
1258:
1246:
1244:
1243:
1238:
1226:
1224:
1223:
1218:
1206:
1204:
1203:
1198:
1196:
1195:
1190:
1182:
1176:
1175:
1170:
1162:
1143:
1141:
1140:
1135:
1130:
1129:
1124:
1116:
1107:
1103:
1102:
1101:
1096:
1088:
1084:
1082:
1081:
1049:
1045:
1044:
1043:
1038:
1030:
1026:
1024:
1023:
997:
993:
992:
991:
986:
978:
974:
972:
971:
945:
944:
939:
931:
909:
907:
906:
901:
896:
892:
880:
876:
872:
871:
866:
858:
852:
851:
846:
838:
832:
831:
826:
818:
811:
807:
795:
791:
787:
786:
781:
773:
767:
766:
761:
753:
747:
746:
741:
733:
726:
722:
710:
706:
702:
701:
696:
688:
682:
681:
676:
668:
662:
661:
656:
648:
632:
630:
629:
624:
619:
618:
613:
605:
599:
598:
593:
585:
579:
578:
573:
565:
559:
558:
553:
545:
528:
526:
525:
520:
518:
510:
505:
497:
492:
484:
475:
473:
472:
467:
448:
446:
445:
440:
432:
431:
426:
418:
412:
411:
406:
398:
392:
391:
386:
378:
365:
363:
362:
357:
337:
335:
334:
329:
314:
312:
311:
306:
291:
289:
288:
283:
268:
266:
265:
260:
111:
110:
107:
106:
103:
100:
97:
67:
53:
39:
26:
4804:
4803:
4799:
4798:
4797:
4795:
4794:
4793:
4779:
4778:
4760:
4759:
4738:
4735:Wayback Machine
4714:
4656:
4651:
4650:
4639:
4635:
4593:
4592:
4588:
4571:
4566:
4565:
4561:
4549:
4545:
4530:10.2307/3620241
4515:
4514:
4510:
4496:
4492:
4481:
4474:
4465:
4461:
4448:
4441:
4432:
4417:
4377:
4376:
4369:
4325:
4324:
4320:
4313:
4298:
4297:
4293:
4259:Euler, Leonhard
4257:
4256:
4252:
4247:: i–xxv, 1–295.
4238:
4237:
4228:
4223:
4207:
4183:
4182:
4163:
4162:
4134:
4133:
4114:
4113:
4094:
4093:
4071:
4070:
4051:
4050:
4031:
4030:
4011:
4010:
3991:
3990:
3968:
3967:
3948:
3947:
3928:
3927:
3909:
3881:
3875:
3836:
3831:
3829:Generalizations
3801:
3794:
3787:
3780:
3765:
3749:
3729:
3701:
3696:
3684:
3640:
3639:
3638:if and only if
3605:
3595:
3585:
3569:
3556:
3543:
3542:
3525:
3515:
3502:
3492:
3479:
3469:
3468:
3450:
3445:
3444:
3403:
3393:
3383:
3373:
3362:
3361:
3339:
3329:
3316:
3306:
3293:
3283:
3270:
3260:
3247:
3237:
3224:
3214:
3209:
3208:
3184:
3179:
3178:
3154:
3149:
3148:
3124:
3119:
3118:
3095:
3090:
3089:
3082:
3049:
3048:
2933:
2932:
2931:has trilinears
2901:
2900:
2776:
2775:
2774:has trilinears
2744:
2743:
2628:
2627:
2626:has trilinears
2596:
2595:
2543:
2542:
2541:has trilinears
2399:
2398:
2277:
2276:
2227:
2226:
2219:
2096:
2095:
2061:
2060:
1900:
1899:
1865:
1860:
1828:
1815:
1802:
1774:
1758:
1750:
1749:
1719:
1706:
1693:
1677:
1661:
1653:
1652:
1619:
1606:
1593:
1565:
1552:
1544:
1543:
1406:
1405:
1347:
1346:
1308:
1307:
1289:
1284:
1249:
1248:
1229:
1228:
1209:
1208:
1183:
1163:
1149:
1148:
1147:In conclusion,
1117:
1089:
1063:
1059:
1031:
1005:
1001:
979:
953:
949:
932:
918:
917:
859:
839:
819:
774:
754:
734:
689:
669:
649:
641:
640:
606:
586:
566:
546:
538:
537:
529:. Further, the
478:
477:
458:
457:
419:
399:
379:
371:
370:
348:
347:
320:
319:
297:
296:
274:
273:
245:
244:
241:
236:
203:. However, the
193:Schiffler point
178:triangle center
170:
165:
94:
90:
73:
65:
63:
51:
49:
37:
35:
24:
17:
12:
11:
5:
4802:
4800:
4792:
4791:
4781:
4780:
4777:
4776:
4757:
4727:
4712:
4693:
4676:
4663:
4655:
4654:External links
4652:
4649:
4648:
4633:
4604:(4): 815–836,
4586:
4559:
4543:
4508:
4490:
4472:
4459:
4439:
4415:
4367:
4336:(1–2): 41–50,
4318:
4312:978-0883850992
4311:
4291:
4250:
4225:
4224:
4222:
4219:
4206:
4203:
4190:
4170:
4147:
4144:
4141:
4121:
4101:
4078:
4069:or a point on
4058:
4038:
4018:
3998:
3975:
3955:
3935:
3908:
3905:
3877:Main article:
3874:
3871:
3835:
3832:
3830:
3827:
3799:
3792:
3785:
3778:
3764:
3761:
3748:
3745:
3728:
3725:
3705:right triangle
3700:
3699:Right triangle
3697:
3695:
3692:
3683:
3680:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3632:
3631:
3620:
3612:
3608:
3602:
3598:
3592:
3588:
3584:
3581:
3576:
3572:
3568:
3563:
3559:
3555:
3550:
3546:
3540:
3537:
3532:
3528:
3522:
3518:
3514:
3509:
3505:
3499:
3495:
3491:
3486:
3482:
3476:
3472:
3465:
3462:
3457:
3453:
3438:
3437:
3436:
3435:
3424:
3421:
3418:
3415:
3410:
3406:
3400:
3396:
3390:
3386:
3380:
3376:
3372:
3369:
3346:
3342:
3336:
3332:
3328:
3323:
3319:
3313:
3309:
3305:
3300:
3296:
3290:
3286:
3282:
3277:
3273:
3267:
3263:
3259:
3254:
3250:
3244:
3240:
3236:
3231:
3227:
3221:
3217:
3191:
3187:
3166:
3161:
3157:
3136:
3131:
3127:
3107:
3102:
3098:
3081:
3078:
3077:
3076:
3065:
3062:
3059:
3056:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
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2949:
2946:
2943:
2940:
2925:
2914:
2911:
2908:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
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2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
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2813:
2810:
2807:
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2801:
2798:
2795:
2792:
2789:
2786:
2783:
2768:
2757:
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2751:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2620:
2609:
2606:
2603:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2520:
2519:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2218:
2215:
2214:
2213:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2080:
2077:
2074:
2071:
2068:
2054:
2053:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1864:
1861:
1859:
1858:Representation
1856:
1855:
1854:
1843:
1840:
1835:
1831:
1827:
1822:
1818:
1814:
1809:
1805:
1801:
1795:
1792:
1786:
1781:
1777:
1773:
1770:
1765:
1761:
1757:
1746:
1745:
1734:
1731:
1726:
1722:
1718:
1713:
1709:
1705:
1700:
1696:
1692:
1689:
1684:
1680:
1676:
1673:
1668:
1664:
1660:
1646:
1645:
1634:
1631:
1626:
1622:
1618:
1613:
1609:
1605:
1600:
1596:
1592:
1586:
1583:
1577:
1572:
1568:
1564:
1559:
1555:
1551:
1487:
1486:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1428:
1425:
1422:
1419:
1416:
1413:
1384:
1383:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1344:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1288:
1285:
1283:
1280:
1256:
1236:
1216:
1193:
1189:
1186:
1179:
1173:
1169:
1166:
1159:
1156:
1145:
1144:
1133:
1127:
1123:
1120:
1113:
1110:
1106:
1099:
1095:
1092:
1080:
1077:
1074:
1067:
1062:
1058:
1055:
1052:
1048:
1041:
1037:
1034:
1022:
1019:
1016:
1009:
1004:
1000:
996:
989:
985:
982:
970:
967:
964:
957:
952:
948:
942:
938:
935:
928:
925:
911:
910:
899:
889:
886:
883:
869:
865:
862:
855:
849:
845:
842:
835:
829:
825:
822:
814:
804:
801:
798:
784:
780:
777:
770:
764:
760:
757:
750:
744:
740:
737:
729:
719:
716:
713:
699:
695:
692:
685:
679:
675:
672:
665:
659:
655:
652:
634:
633:
622:
616:
612:
609:
602:
596:
592:
589:
582:
576:
572:
569:
562:
556:
552:
549:
516:
513:
508:
503:
500:
495:
490:
487:
465:
450:
449:
438:
435:
429:
425:
422:
415:
409:
405:
402:
395:
389:
385:
382:
355:
327:
304:
281:
258:
255:
252:
240:
239:A vector proof
237:
235:
232:
169:
166:
164:
161:
87:Leonhard Euler
85:, named after
64:
50:
36:
23:
15:
13:
10:
9:
6:
4:
3:
2:
4801:
4790:
4787:
4786:
4784:
4772:
4771:
4766:
4763:
4758:
4755:
4751:
4750:
4745:
4741:
4736:
4732:
4728:
4725:
4721:
4717:
4713:
4711:
4710:
4705:
4701:
4697:
4694:
4692:
4688:
4684:
4680:
4677:
4675:
4671:
4667:
4664:
4661:
4658:
4657:
4653:
4646:
4644:
4637:
4634:
4629:
4625:
4621:
4617:
4612:
4607:
4603:
4599:
4598:
4590:
4587:
4581:
4577:
4570:
4563:
4560:
4557:
4553:
4547:
4544:
4539:
4535:
4531:
4527:
4523:
4519:
4512:
4509:
4506:
4502:
4501:
4494:
4491:
4488:
4485:
4479:
4477:
4473:
4469:
4463:
4460:
4456:
4455:0-486-61348-8
4452:
4446:
4444:
4440:
4436:
4430:
4428:
4426:
4424:
4422:
4420:
4416:
4411:
4407:
4403:
4399:
4395:
4391:
4387:
4383:
4382:
4374:
4372:
4368:
4363:
4359:
4355:
4351:
4347:
4343:
4339:
4335:
4331:
4330:
4322:
4319:
4314:
4308:
4304:
4303:
4295:
4292:
4289:
4285:
4282:
4278:
4275:Reprinted in
4272:
4268:
4264:
4260:
4254:
4251:
4246:
4242:
4235:
4233:
4231:
4227:
4220:
4218:
4216:
4212:
4204:
4202:
4188:
4168:
4159:
4145:
4142:
4139:
4119:
4099:
4090:
4076:
4056:
4036:
4016:
3996:
3987:
3973:
3953:
3933:
3926:Suppose that
3924:
3922:
3918:
3914:
3906:
3904:
3902:
3898:
3894:
3890:
3886:
3880:
3872:
3870:
3868:
3864:
3860:
3856:
3853:
3849:
3845:
3841:
3834:Quadrilateral
3833:
3828:
3826:
3824:
3820:
3816:
3811:
3809:
3805:
3798:
3791:
3784:
3777:
3774:
3770:
3762:
3760:
3758:
3754:
3746:
3744:
3742:
3738:
3734:
3726:
3724:
3722:
3718:
3714:
3710:
3706:
3698:
3693:
3691:
3689:
3681:
3679:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3637:
3618:
3610:
3606:
3600:
3596:
3590:
3586:
3582:
3579:
3574:
3570:
3566:
3561:
3557:
3553:
3548:
3544:
3538:
3535:
3530:
3526:
3520:
3516:
3512:
3507:
3503:
3497:
3493:
3489:
3484:
3480:
3474:
3470:
3463:
3460:
3455:
3451:
3443:
3442:
3441:
3422:
3419:
3416:
3413:
3408:
3404:
3398:
3394:
3388:
3384:
3378:
3374:
3370:
3367:
3360:
3359:
3344:
3340:
3334:
3330:
3326:
3321:
3317:
3311:
3307:
3303:
3298:
3294:
3288:
3284:
3280:
3275:
3271:
3265:
3261:
3257:
3252:
3248:
3242:
3238:
3234:
3229:
3225:
3219:
3215:
3207:
3206:
3205:
3189:
3185:
3164:
3159:
3155:
3134:
3129:
3125:
3105:
3100:
3096:
3087:
3079:
3063:
3060:
3057:
3054:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2930:
2926:
2912:
2909:
2906:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2773:
2769:
2755:
2752:
2749:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2625:
2621:
2607:
2604:
2601:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2540:
2536:
2535:
2534:
2533:For example:
2531:
2529:
2525:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2397:
2396:
2395:
2381:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2224:
2216:
2200:
2197:
2194:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2164:
2161:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2131:
2128:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2094:
2093:
2092:
2078:
2075:
2072:
2069:
2066:
2059:
2040:
2037:
2034:
2028:
2025:
2022:
2016:
2013:
2007:
2004:
1998:
1995:
1992:
1989:
1983:
1980:
1977:
1971:
1968:
1962:
1959:
1953:
1950:
1947:
1944:
1938:
1935:
1932:
1926:
1923:
1917:
1914:
1908:
1905:
1898:
1897:
1896:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1862:
1857:
1841:
1833:
1829:
1825:
1820:
1816:
1812:
1807:
1803:
1793:
1790:
1784:
1779:
1775:
1771:
1768:
1763:
1759:
1755:
1748:
1747:
1732:
1724:
1720:
1716:
1711:
1707:
1703:
1698:
1694:
1687:
1682:
1678:
1674:
1671:
1666:
1662:
1658:
1651:
1650:
1649:
1648:In addition,
1632:
1624:
1620:
1616:
1611:
1607:
1603:
1598:
1594:
1584:
1581:
1575:
1570:
1566:
1562:
1557:
1553:
1549:
1542:
1541:
1540:
1538:
1534:
1530:
1526:
1523:
1518:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1426:
1423:
1420:
1417:
1414:
1411:
1404:
1403:
1402:
1400:
1395:
1393:
1389:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1345:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1306:
1305:
1304:
1302:
1298:
1294:
1286:
1281:
1279:
1277:
1273:
1268:
1254:
1234:
1214:
1187:
1184:
1177:
1167:
1164:
1157:
1154:
1131:
1121:
1118:
1111:
1108:
1104:
1093:
1090:
1060:
1056:
1053:
1050:
1046:
1035:
1032:
1002:
998:
994:
983:
980:
950:
946:
936:
933:
926:
923:
916:
915:
914:
897:
887:
884:
881:
863:
860:
853:
843:
840:
833:
823:
820:
812:
802:
799:
796:
778:
775:
768:
758:
755:
748:
738:
735:
727:
717:
714:
711:
693:
690:
683:
673:
670:
663:
653:
650:
639:
638:
637:
620:
610:
607:
600:
590:
587:
580:
570:
567:
560:
550:
547:
536:
535:
534:
532:
514:
511:
506:
501:
498:
493:
488:
485:
463:
455:
436:
433:
423:
420:
413:
403:
400:
393:
383:
380:
369:
368:
367:
353:
345:
341:
325:
318:
302:
295:
279:
272:
256:
253:
250:
238:
233:
231:
229:
225:
221:
217:
212:
210:
206:
202:
198:
194:
190:
185:
183:
179:
175:
167:
162:
160:
158:
154:
153:quadrilateral
149:
147:
143:
139:
135:
131:
127:
123:
119:
115:
109:
88:
84:
80:
71:
61:
57:
47:
43:
34:
30:
21:
4768:
4765:"Euler Line"
4747:
4731:Ghostarchive
4729:Archived at
4723:
4709:Cut-the-Knot
4707:
4666:"Euler Line"
4642:
4636:
4601:
4595:
4589:
4579:
4575:
4562:
4551:
4546:
4521:
4517:
4511:
4498:
4493:
4483:
4467:
4462:
4434:
4385:
4379:
4361:
4333:
4327:
4321:
4301:
4294:
4276:
4270:
4266:
4253:
4244:
4240:
4208:
4160:
4091:
3988:
3925:
3910:
3882:
3866:
3862:
3854:
3847:
3843:
3837:
3812:
3807:
3796:
3789:
3782:
3775:
3768:
3766:
3750:
3730:
3702:
3685:
3635:
3633:
3439:
3083:
2539:circumcenter
2532:
2527:
2522:formed as a
2521:
2222:
2220:
2055:
1888:
1884:
1880:
1876:
1872:
1868:
1866:
1647:
1536:
1532:
1528:
1524:
1522:circumradius
1519:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1488:
1398:
1396:
1387:
1386:The segment
1385:
1300:
1296:
1292:
1290:
1271:
1269:
1146:
912:
635:
451:
344:free vectors
339:
271:circumcenter
242:
220:circumcircle
213:
197:Exeter point
186:
171:
150:
142:Exeter point
134:circumcenter
126:central line
120:that is not
82:
76:
70:circumcenter
4749:Numberphile
4277:Opera Omnia
3897:Monge point
3885:tetrahedron
3873:Tetrahedron
3819:orthocenter
3755:(one whose
1397:The center
317:orthocenter
157:tetrahedron
130:orthocenter
122:equilateral
60:orthocenter
4221:References
4049:is either
3850:, and the
3804:concurrent
3790:A, B, C, F
3713:hypotenuse
1282:Properties
1272:Euler line
342:relies on
199:, and the
124:. It is a
83:Euler line
29:the center
4770:MathWorld
4611:1301.0496
4582:: 289–295
4410:125341434
4358:121434528
4215:directrix
3917:simplices
3859:collinear
3717:altitudes
3658:
3649:
3464:−
3061:−
3029:
3020:
3014:−
3008:
2996:
2987:
2981:−
2975:
2963:
2954:
2948:−
2942:
2881:
2872:
2857:
2845:
2836:
2821:
2809:
2800:
2785:
2724:
2715:
2703:
2691:
2682:
2670:
2658:
2649:
2637:
2576:
2564:
2552:
2504:
2495:
2480:
2468:
2459:
2444:
2432:
2423:
2408:
2373:
2364:
2352:
2343:
2331:
2322:
2310:
2298:
2286:
2260:
2248:
2236:
2195:γ
2186:
2180:−
2174:
2162:β
2153:
2147:−
2141:
2129:α
2120:
2114:−
2108:
2079:γ
2073:β
2067:α
2026:−
2017:
1999:
1981:−
1972:
1954:
1936:−
1927:
1909:
1785:−
1688:−
1576:−
1443:⋅
1192:→
1172:→
1158:⋅
1126:→
1112:−
1098:→
1066:∑
1057:−
1040:→
1008:∑
988:→
956:∑
941:→
927:⋅
868:→
848:→
828:→
783:→
763:→
743:→
698:→
678:→
658:→
615:→
595:→
575:→
555:→
533:reads as
428:→
408:→
388:→
340:collinear
174:collinear
56:Altitudes
4783:Category
4733:and the
4628:12307207
4402:40378417
4261:(1767).
4211:extended
3741:incenter
2624:centroid
1887: :
1883: :
1863:Equation
1274:and the
315:and the
294:centroid
205:incenter
155:and the
138:centroid
118:triangle
112:), is a
79:geometry
46:centroid
4754:YouTube
4702:" and "
4672:at the
4538:3620241
4350:2430410
4284:0061061
4132:, then
4029:, then
3757:medians
3711:to the
226:of the
42:Medians
31:of the
4685:, and
4626:
4536:
4453:
4408:
4400:
4356:
4348:
4309:
4092:2. If
3989:1. If
3709:median
1535:, and
292:, the
234:Proofs
228:orthic
195:, the
191:, the
180:, the
140:, the
136:, the
132:, the
81:, the
66:
54:
52:
40:
38:
25:
4624:S2CID
4606:arXiv
4572:(PDF)
4534:JSTOR
4406:S2CID
4398:JSTOR
4354:S2CID
3893:faces
3887:is a
3838:In a
3771:with
3703:In a
3084:In a
3080:Slope
4668:and
4451:ISBN
4307:ISBN
3857:are
3802:are
3795:and
3781:and
3147:and
2927:The
2770:The
2622:The
2537:The
1867:Let
1509:at 6
1497:at 2
1247:and
476:are
338:are
243:Let
214:The
114:line
4706:",
4698:, "
4689:at
4616:doi
4526:doi
4390:doi
4338:doi
4245:129
3865:= 2
3808:ABC
3769:ABC
3655:tan
3646:tan
3026:cos
3017:cos
3005:cos
2993:cos
2984:cos
2972:cos
2960:cos
2951:cos
2939:cos
2878:cos
2869:cos
2854:cos
2842:cos
2833:cos
2818:cos
2806:cos
2797:cos
2782:cos
2721:cos
2712:cos
2700:cos
2688:cos
2679:cos
2667:cos
2655:cos
2646:cos
2634:cos
2573:cos
2561:cos
2549:cos
2501:cos
2492:cos
2477:cos
2465:cos
2456:cos
2441:cos
2429:cos
2420:cos
2405:cos
2370:cos
2361:cos
2349:cos
2340:cos
2328:cos
2319:cos
2307:sec
2295:sec
2283:sec
2257:cos
2245:cos
2233:cos
2183:tan
2171:tan
2150:tan
2138:tan
2117:tan
2105:tan
2091:is
2014:sin
1996:sin
1969:sin
1951:sin
1924:sin
1906:sin
456:of
77:In
4785::
4767:.
4752:,
4746:,
4737::
4722:,
4718:,
4681:,
4622:,
4614:,
4602:51
4600:,
4578:,
4574:,
4532:,
4522:75
4520:,
4475:^
4442:^
4418:^
4404:,
4396:,
4386:91
4384:,
4370:^
4360:,
4352:,
4346:MR
4344:,
4334:52
4332:,
4281:MR
4271:11
4269:.
4243:.
4229:^
4217:.
4158:.
4089:.
3911:A
3883:A
3869:.
3867:GO
3863:HG
3810:.
3667:3.
3636:BC
3423:0.
3064:1.
2913:2.
2756:1.
2608:0.
2530:.
2201:0.
2041:0.
1875:,
1871:,
1539::
1531:,
1517:.
1394:.
1388:GH
1227:,
437:0.
159:.
105:ər
99:ɔɪ
4773:.
4662:.
4631:.
4618::
4608::
4584:.
4580:6
4541:.
4528::
4413:.
4392::
4365:.
4340::
4315:.
4189:E
4169:P
4146:C
4143:=
4140:E
4120:C
4100:P
4077:L
4057:L
4037:E
4017:L
3997:P
3974:P
3954:E
3934:P
3855:O
3848:G
3844:H
3800:2
3797:F
3793:1
3786:2
3783:F
3779:1
3776:F
3664:=
3661:C
3652:B
3619:.
3611:3
3607:m
3601:2
3597:m
3591:1
3587:m
3583:3
3580:+
3575:3
3571:m
3567:+
3562:2
3558:m
3554:+
3549:1
3545:m
3539:3
3536:+
3531:3
3527:m
3521:2
3517:m
3513:+
3508:3
3504:m
3498:1
3494:m
3490:+
3485:2
3481:m
3475:1
3471:m
3461:=
3456:E
3452:m
3420:=
3417:3
3414:+
3409:E
3405:m
3399:3
3395:m
3389:2
3385:m
3379:1
3375:m
3371:3
3368:+
3345:E
3341:m
3335:3
3331:m
3327:+
3322:E
3318:m
3312:2
3308:m
3304:+
3299:3
3295:m
3289:2
3285:m
3281:+
3276:E
3272:m
3266:1
3262:m
3258:+
3253:3
3249:m
3243:1
3239:m
3235:+
3230:2
3226:m
3220:1
3216:m
3190:E
3186:m
3165:,
3160:3
3156:m
3135:,
3130:2
3126:m
3106:,
3101:1
3097:m
3058:=
3055:t
3035:,
3032:B
3023:A
3011:C
3002::
2999:A
2990:C
2978:B
2969::
2966:C
2957:B
2945:A
2910:=
2907:t
2887:,
2884:B
2875:A
2866:2
2863:+
2860:C
2851::
2848:A
2839:C
2830:2
2827:+
2824:B
2815::
2812:C
2803:B
2794:2
2791:+
2788:A
2753:=
2750:t
2730:,
2727:B
2718:A
2709:+
2706:C
2697::
2694:A
2685:C
2676:+
2673:B
2664::
2661:C
2652:B
2643:+
2640:A
2605:=
2602:t
2582:,
2579:C
2570::
2567:B
2558::
2555:A
2528:t
2507:B
2498:A
2489:t
2486:+
2483:C
2474::
2471:A
2462:C
2453:t
2450:+
2447:B
2438::
2435:C
2426:B
2417:t
2414:+
2411:A
2382:,
2379:)
2376:B
2367:A
2358::
2355:A
2346:C
2337::
2334:C
2325:B
2316:=
2313:C
2304::
2301:B
2292::
2289:A
2263:C
2254::
2251:B
2242::
2239:A
2223:t
2198:=
2192:)
2189:A
2177:B
2168:(
2165:+
2159:)
2156:C
2144:A
2135:(
2132:+
2126:)
2123:B
2111:C
2102:(
2076::
2070::
2038:=
2035:z
2032:)
2029:B
2023:A
2020:(
2011:)
2008:C
2005:2
2002:(
1993:+
1990:y
1987:)
1984:A
1978:C
1975:(
1966:)
1963:B
1960:2
1957:(
1948:+
1945:x
1942:)
1939:C
1933:B
1930:(
1921:)
1918:A
1915:2
1912:(
1889:z
1885:y
1881:x
1877:C
1873:B
1869:A
1842:.
1839:)
1834:2
1830:c
1826:+
1821:2
1817:b
1813:+
1808:2
1804:a
1800:(
1794:9
1791:4
1780:2
1776:R
1772:4
1769:=
1764:2
1760:H
1756:G
1733:;
1730:)
1725:2
1721:c
1717:+
1712:2
1708:b
1704:+
1699:2
1695:a
1691:(
1683:2
1679:R
1675:9
1672:=
1667:2
1663:H
1659:O
1633:.
1630:)
1625:2
1621:c
1617:+
1612:2
1608:b
1604:+
1599:2
1595:a
1591:(
1585:9
1582:1
1571:2
1567:R
1563:=
1558:2
1554:O
1550:G
1537:c
1533:b
1529:a
1525:R
1515:t
1511:t
1507:H
1503:t
1499:t
1495:G
1491:O
1474:.
1471:N
1468:G
1465:3
1462:=
1459:H
1456:N
1452:,
1449:N
1446:G
1440:2
1437:=
1434:G
1431:O
1427:,
1424:H
1421:N
1418:=
1415:N
1412:O
1399:N
1371:.
1368:O
1365:G
1362:3
1359:=
1356:H
1353:O
1332:;
1329:O
1326:G
1323:2
1320:=
1317:H
1314:G
1301:H
1297:O
1293:G
1255:H
1235:G
1215:O
1188:H
1185:O
1178:=
1168:G
1165:O
1155:3
1132:.
1122:H
1119:O
1109:=
1105:)
1094:A
1091:O
1079:c
1076:y
1073:c
1061:(
1054:0
1051:=
1047:)
1036:O
1033:A
1021:c
1018:y
1015:c
1003:(
999:+
995:)
984:A
981:G
969:c
966:y
963:c
951:(
947:=
937:O
934:G
924:3
898:.
893:)
888:O
885:G
882:C
864:O
861:C
854:+
844:C
841:G
834:=
824:O
821:G
813:,
808:)
803:O
800:G
797:B
779:O
776:B
769:+
759:B
756:G
749:=
739:O
736:G
728:,
723:)
718:O
715:G
712:A
694:O
691:A
684:+
674:A
671:G
664:=
654:O
651:G
621:.
611:C
608:O
601:+
591:B
588:O
581:+
571:A
568:O
561:=
551:H
548:O
515:3
512:1
507::
502:3
499:1
494::
489:3
486:1
464:G
434:=
424:C
421:G
414:+
404:B
401:G
394:+
384:A
381:G
354:G
326:H
303:G
280:O
257:C
254:B
251:A
108:/
102:l
96:ˈ
93:/
89:(
72:)
62:)
48:)
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