2482:
2615:
2490:
A point in the interior of a triangle is the center of an inellipse of the triangle if and only if the point lies in the interior of the triangle whose vertices lie at the midpoints of the original triangle's sides. For a given point inside that
2269:
824:
2059:
1441:
556:
2286:
1879:
1124:
1534:
of the one with greatest area coincides with the center of the ellipse. The given ellipse, going through this triangle's three vertices and centered at the triangle's centroid, is called the triangle's
2526:
700:
2291:
720:
1279:
1522:
189:
906:
1743:
1686:
1616:
307:
386:
1168:
245:
2086:
715:
1908:
2763:
1300:
418:
2477:{\displaystyle {\begin{aligned}L&=q_{1}r_{2}-r_{1}q_{2},\\M&=r_{1}p_{2}-p_{1}r_{2},\\N&=p_{1}q_{2}-q_{1}p_{2}.\end{aligned}}}
1756:
2643:
The lines connecting the tangency points of any inellipse of a triangle with the opposite vertices of the triangle are concurrent.
980:
2610:{\displaystyle {\frac {\text{Area of inellipse}}{\text{Area of triangle}}}=\pi {\sqrt {(1-2\alpha )(1-2\beta )(1-2\gamma )}},}
577:
1183:
1462:
2507:
102:
840:
2687:, the unique ellipse that passes through a triangle's three vertices and is centered at the triangle's
1695:
1638:
1562:
253:
2860:
2506:. In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum
1624:
in which case it is tangent externally to one of the sides of the triangle and is tangent to the
344:
2702:
2706:
2684:
2264:{\displaystyle L^{4}x^{2}+M^{4}y^{2}+N^{4}z^{2}-2M^{2}N^{2}yz-2N^{2}L^{2}zx-2L^{2}M^{2}xy=0,}
1536:
1450:
1135:
969:
212:
202:
2696:, the unique conic which passes through a triangle's three vertices, its centroid, and its
8:
2720:
819:{\displaystyle {\begin{aligned}wv+vz&=0,\\uz+wx&=0,\\vx+uy&=0.\end{aligned}}}
398:
in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.
327:
20:
16:
Conic section that passes through the vertices of a triangle or is tangent to its sides
2749:
2743:
2710:
2499:
32:
2825:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.
2746:, the unique ellipse that is tangent to a triangle's three sides at their midpoints
44:
2492:
2838:
2785:
2752:, the unique ellipse tangent to a triangle's sides at the contact points of its
2758:
2693:
2054:{\displaystyle X^{2}=(p_{1}+p_{2}t)^{2}:(q_{1}+q_{2}t)^{2}:(r_{1}+r_{2}t)^{2}.}
2854:
2653:
1625:
1170:
is a point on the general circumconic, then the line tangent to the conic at
48:
28:
2844:
2797:
2727:, and various other notable points, and has center on the nine-point circle.
2714:
2674:
1892:
2784:
Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web
Resource.
2740:, the unique circle that is internally tangent to a triangle's three sides
2724:
2697:
2723:, a rectangular hyperbola that passes through a triangle's orthocenter,
2502:, also called the midpoint inellipse, with its center at the triangle's
2796:
Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web
Resource.
1436:{\displaystyle u^{2}a^{2}+v^{2}b^{2}+w^{2}c^{2}-2vwbc-2wuca-2uvab=0,}
551:{\displaystyle u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy=0.}
2753:
2737:
2688:
2657:
2503:
1550:
1531:
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36:
2713:
and passing through the triangle's three vertices as well as its
2074:
2678:
1874:{\displaystyle X=(p_{1}+p_{2}t):(q_{1}+q_{2}t):(r_{1}+r_{2}t).}
911:
The lines tangent to the general inconic are the sidelines of
2623:
which is maximized by the centroid's barycentric coordinates
705:
The lines tangent to the general circumconic at the vertices
1119:{\displaystyle (cx-az)(ay-bx):(ay-bx)(bz-cy):(bz-cy)(cx-az)}
2821:
Chakerian, G. D. "A Distorted View of
Geometry." Ch. 7 in
2656:
fall on the line segment connecting the midpoints of the
2495:, the inellipse with its center at that point is unique.
953:
Each noncircular circumconic meets the circumcircle of
2529:
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1911:
1759:
1698:
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718:
695:{\displaystyle u(-au+bv+cw):v(au-bv+cw):w(au+bv-cw).}
580:
421:
347:
256:
215:
105:
1530:Of all triangles inscribed in a given ellipse, the
571:The center of the general circumconic is the point
2609:
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2053:
1873:
1737:
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1610:
1516:
1435:
1273:
1162:
1118:
900:
818:
694:
550:
380:
301:
239:
183:
2852:
2717:, orthocenter, and various other notable centers
2681:that passes through a triangle's three vertices
834:The center of the general inconic is the point
73:denotes not only the vertex but also the angle
2647:
2786:http://mathworld.wolfram.com/Circumconic.html
1274:{\displaystyle (vr+wq)x+(wp+ur)y+(uq+vp)z=0.}
561:
2817:
2815:
2813:
2811:
2809:
2807:
2805:
2498:The inellipse with the largest area is the
58:are distinct non-collinear points, and let
1517:{\displaystyle u\cos A+v\cos B+w\cos C=0.}
2798:http://mathworld.wolfram.com/Inconic.html
2652:All the centers of inellipses of a given
2802:
65:denote the triangle whose vertices are
2853:
1287:The general circumconic reduces to a
405:is tangent to the three sidelines of
184:{\displaystyle a=|BC|,b=|CA|,c=|AB|,}
391:This line meets the circumcircle of
13:
901:{\displaystyle cv+bw:aw+cu:bu+av.}
14:
2872:
2832:
1738:{\displaystyle p_{2}:q_{2}:r_{2}}
1681:{\displaystyle p_{1}:q_{1}:r_{1}}
1626:extensions of the other two sides
1549:The general inconic reduces to a
942:
209:is the locus of a variable point
947:
2073:is the inconic, necessarily an
334:on the circumconic, other than
2790:
2778:
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2548:
2039:
2009:
1997:
1967:
1955:
1925:
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1830:
1801:
1795:
1766:
1611:{\displaystyle ubc+vca+wab=0,}
1259:
1241:
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1205:
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1005:
1002:
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686:
659:
650:
623:
614:
584:
566:
302:{\displaystyle uyz+vzx+wxy=0,}
174:
163:
149:
138:
124:
113:
69:. Following common practice,
31:that passes through the three
1:
2771:
2517:of the inellipse's center, is
412:and is given by the equation
1747:are distinct points, and let
1543:
966:fourth point of intersection
7:
2663:
2648:Extension to quadrilaterals
381:{\displaystyle ux+vy+wz=0.}
10:
2877:
829:
2709:centered on a triangle's
918:, given by the equations
562:Centers and tangent lines
338:, is a point on the line
2508:barycentric coordinates
2077:, given by the equation
1163:{\displaystyle P=p:q:r}
247:satisfying an equation
240:{\displaystyle X=x:y:z}
47:in the sides, possibly
2660:of the quadrilateral.
2611:
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2265:
2055:
1875:
1739:
1682:
1612:
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1164:
1120:
960:in a point other than
902:
820:
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552:
382:
303:
241:
185:
2707:rectangular hyperbola
2685:Steiner circumellipse
2612:
2479:
2266:
2056:
1876:
1740:
1683:
1613:
1537:Steiner circumellipse
1519:
1451:rectangular hyperbola
1438:
1276:
1165:
1121:
970:trilinear coordinates
903:
821:
697:
553:
383:
304:
242:
203:trilinear coordinates
186:
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1136:
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716:
578:
419:
345:
254:
213:
103:
84:, and similarly for
2721:Feuerbach hyperbola
1891:ranges through the
964:, often called the
709:are, respectively,
207:general circumconic
191:the sidelengths of
43:is a conic section
2823:Mathematical Plums
2607:
2474:
2472:
2261:
2051:
1899:is a line. Define
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1735:
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1608:
1514:
1433:
1271:
1160:
1116:
898:
816:
814:
692:
548:
378:
328:isogonal conjugate
299:
237:
181:
21:Euclidean geometry
2750:Mandart inellipse
2744:Steiner inellipse
2711:nine-point circle
2694:Kiepert hyperbola
2602:
2538:
2537:
2534:
2533:Area of inellipse
2500:Steiner inellipse
1887:As the parameter
51:, of a triangle.
2868:
2826:
2819:
2800:
2794:
2788:
2782:
2759:Kiepert parabola
2637:
2616:
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2547:
2539:
2536:Area of triangle
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2144:
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2121:
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2108:
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2098:
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2060:
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1962:
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1949:
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1936:
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1920:
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1890:
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1877:
1872:
1861:
1860:
1848:
1847:
1826:
1825:
1813:
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1791:
1790:
1778:
1777:
1746:
1744:
1742:
1741:
1736:
1734:
1733:
1721:
1720:
1708:
1707:
1689:
1687:
1685:
1684:
1679:
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1676:
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1650:
1617:
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1614:
1609:
1523:
1521:
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1515:
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1369:
1368:
1359:
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1336:
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1322:
1313:
1312:
1280:
1278:
1277:
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1169:
1167:
1166:
1161:
1125:
1123:
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1117:
963:
959:
938:
931:
924:
917:
907:
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904:
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822:
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693:
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487:
486:
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464:
463:
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411:
397:
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379:
337:
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246:
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197:
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2779:
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2493:medial triangle
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2025:
2016:
2012:
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1996:
1987:
1983:
1974:
1970:
1958:
1954:
1945:
1941:
1932:
1928:
1916:
1912:
1910:
1907:
1906:
1896:
1895:, the locus of
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1821:
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1133:
982:
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978:
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954:
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945:
933:
926:
919:
912:
842:
839:
838:
832:
813:
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784:
783:
770:
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751:
738:
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713:
706:
579:
576:
575:
569:
564:
482:
478:
472:
468:
459:
455:
449:
445:
436:
432:
426:
422:
420:
417:
416:
406:
403:general inconic
392:
346:
343:
342:
335:
331:
313:
312:for some point
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100:
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2861:Conic sections
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2833:External links
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943:Other features
941:
909:
908:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
831:
828:
827:
826:
811:
808:
805:
803:
801:
798:
795:
792:
789:
786:
785:
782:
779:
776:
773:
771:
769:
766:
763:
760:
757:
754:
753:
750:
747:
744:
741:
739:
737:
734:
731:
728:
725:
722:
721:
703:
702:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
568:
565:
563:
560:
559:
558:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
485:
481:
475:
471:
467:
462:
458:
452:
448:
444:
439:
435:
429:
425:
389:
388:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
330:of each point
310:
309:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
236:
233:
230:
227:
224:
221:
218:
180:
176:
172:
169:
165:
161:
158:
155:
151:
147:
144:
140:
136:
133:
130:
126:
122:
119:
115:
111:
108:
15:
9:
6:
4:
3:
2:
2873:
2862:
2859:
2858:
2856:
2846:
2843:
2840:
2837:
2836:
2824:
2818:
2816:
2814:
2812:
2810:
2808:
2806:
2799:
2793:
2787:
2781:
2777:
2765:
2762:
2760:
2757:
2755:
2751:
2748:
2745:
2742:
2739:
2736:
2735:
2734:
2731:
2726:
2722:
2719:
2716:
2712:
2708:
2705:hyperbola, a
2704:
2701:
2699:
2695:
2692:
2690:
2686:
2683:
2680:
2677:, the unique
2676:
2673:
2672:
2671:
2668:
2667:
2661:
2659:
2655:
2654:quadrilateral
2642:
2641:
2635:
2631:
2627:
2622:
2621:
2604:
2596:
2593:
2590:
2587:
2578:
2575:
2572:
2569:
2560:
2557:
2554:
2551:
2543:
2540:
2523:
2522:
2521:
2520:
2514:
2509:
2505:
2501:
2497:
2494:
2489:
2488:
2467:
2462:
2458:
2452:
2448:
2444:
2439:
2435:
2429:
2425:
2421:
2419:
2414:
2407:
2402:
2398:
2392:
2388:
2384:
2379:
2375:
2369:
2365:
2361:
2359:
2354:
2347:
2342:
2338:
2332:
2328:
2324:
2319:
2315:
2309:
2305:
2301:
2299:
2294:
2283:
2282:
2281:
2280:
2276:
2275:
2258:
2255:
2252:
2249:
2246:
2241:
2237:
2231:
2227:
2223:
2220:
2217:
2214:
2209:
2205:
2199:
2195:
2191:
2188:
2185:
2182:
2177:
2173:
2167:
2163:
2159:
2156:
2151:
2147:
2141:
2137:
2133:
2128:
2124:
2118:
2114:
2110:
2105:
2101:
2095:
2091:
2083:
2082:
2081:
2080:
2076:
2071:
2067:The locus of
2066:
2065:
2048:
2043:
2035:
2030:
2026:
2022:
2017:
2013:
2006:
2001:
1993:
1988:
1984:
1980:
1975:
1971:
1964:
1959:
1951:
1946:
1942:
1938:
1933:
1929:
1922:
1917:
1913:
1905:
1904:
1903:
1902:
1894:
1886:
1885:
1868:
1862:
1857:
1853:
1849:
1844:
1840:
1833:
1827:
1822:
1818:
1814:
1809:
1805:
1798:
1792:
1787:
1783:
1779:
1774:
1770:
1763:
1760:
1753:
1752:
1751:
1750:
1730:
1726:
1722:
1717:
1713:
1709:
1704:
1700:
1673:
1669:
1665:
1660:
1656:
1652:
1647:
1643:
1633:Suppose that
1632:
1631:
1627:
1623:
1622:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1559:
1558:
1557:
1556:
1552:
1548:
1547:
1538:
1533:
1529:
1528:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1459:
1458:
1457:
1456:
1452:
1448:
1447:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1365:
1361:
1355:
1351:
1347:
1342:
1338:
1332:
1328:
1324:
1319:
1315:
1309:
1305:
1297:
1296:
1295:
1294:
1290:
1286:
1285:
1268:
1265:
1262:
1256:
1253:
1250:
1247:
1244:
1238:
1235:
1229:
1226:
1223:
1220:
1217:
1211:
1208:
1202:
1199:
1196:
1193:
1190:
1180:
1179:
1178:
1177:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1131:
1130:
1110:
1107:
1104:
1101:
1098:
1089:
1086:
1083:
1080:
1077:
1071:
1065:
1062:
1059:
1056:
1053:
1044:
1041:
1038:
1035:
1032:
1026:
1020:
1017:
1014:
1011:
1008:
999:
996:
993:
990:
987:
977:
976:
975:
974:
971:
967:
958:
952:
951:
940:
936:
929:
922:
916:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
837:
836:
835:
809:
806:
804:
799:
796:
793:
790:
787:
780:
777:
774:
772:
767:
764:
761:
758:
755:
748:
745:
742:
740:
735:
732:
729:
726:
723:
712:
711:
710:
689:
683:
680:
677:
674:
671:
668:
665:
662:
656:
653:
647:
644:
641:
638:
635:
632:
629:
626:
620:
617:
611:
608:
605:
602:
599:
596:
593:
590:
587:
581:
574:
573:
572:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
483:
479:
473:
469:
465:
460:
456:
450:
446:
442:
437:
433:
427:
423:
415:
414:
413:
410:
404:
399:
396:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
341:
340:
339:
329:
324:
320:
316:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
250:
249:
248:
234:
231:
228:
225:
222:
219:
216:
208:
204:
199:
196:
178:
170:
167:
159:
156:
153:
145:
142:
134:
131:
128:
120:
117:
109:
106:
97:
92:as angles in
78:
63:
52:
50:
46:
42:
38:
34:
30:
29:conic section
26:
22:
2847:at MathWorld
2841:at MathWorld
2822:
2792:
2780:
2764:Yff parabola
2732:
2715:circumcenter
2675:Circumcircle
2670:Circumconics
2669:
2651:
2633:
2629:
2625:
2512:
2069:
1893:real numbers
965:
956:
934:
927:
920:
914:
910:
833:
704:
570:
408:
402:
400:
394:
390:
322:
318:
314:
311:
206:
200:
194:
95:
76:
61:
53:
40:
24:
18:
2839:Circumconic
2725:Nagel point
2698:orthocenter
1174:is given by
968:, given by
948:Circumconic
567:Circumconic
25:circumconic
2772:References
80:at vertex
2754:excircles
2658:diagonals
2597:γ
2591:−
2579:β
2573:−
2561:α
2555:−
2544:π
2445:−
2385:−
2325:−
2221:−
2189:−
2157:−
1503:
1488:
1473:
1449:and to a
1407:−
1389:−
1371:−
1105:−
1084:−
1060:−
1039:−
1015:−
994:−
678:−
633:−
588:−
525:−
507:−
489:−
45:inscribed
39:, and an
2855:Category
2738:Incircle
2733:Inconics
2689:centroid
2664:Examples
2504:centroid
1551:parabola
1532:centroid
1289:parabola
321: :
317: :
54:Suppose
49:extended
37:triangle
33:vertices
2845:Inconic
2703:Jeřábek
2513:α, β, γ
2075:ellipse
1745:
1692:
1688:
1635:
1544:Inconic
962:A, B, C
830:Inconic
707:A, B, C
336:A, B, C
326:. The
99:. Let
67:A, B, C
56:A, B, C
41:inconic
2679:circle
205:, the
2277:where
35:of a
27:is a
1690:and
401:The
88:and
23:, a
2636:= ⅓
1500:cos
1485:cos
1470:cos
1132:If
957:ABC
937:= 0
930:= 0
923:= 0
915:ABC
409:ABC
395:ABC
201:In
195:ABC
96:ABC
77:BAC
62:ABC
19:In
2857::
2804:^
2632:=
2628:=
1512:0.
1269:0.
939:.
932:,
925:,
810:0.
546:0.
376:0.
198:.
2638:.
2634:γ
2630:β
2626:α
2605:,
2600:)
2594:2
2588:1
2585:(
2582:)
2576:2
2570:1
2567:(
2564:)
2558:2
2552:1
2549:(
2541:=
2515:)
2511:(
2468:.
2463:2
2459:p
2453:1
2449:q
2440:2
2436:q
2430:1
2426:p
2422:=
2415:N
2408:,
2403:2
2399:r
2393:1
2389:p
2380:2
2376:p
2370:1
2366:r
2362:=
2355:M
2348:,
2343:2
2339:q
2333:1
2329:r
2320:2
2316:r
2310:1
2306:q
2302:=
2295:L
2259:,
2256:0
2253:=
2250:y
2247:x
2242:2
2238:M
2232:2
2228:L
2224:2
2218:x
2215:z
2210:2
2206:L
2200:2
2196:N
2192:2
2186:z
2183:y
2178:2
2174:N
2168:2
2164:M
2160:2
2152:2
2148:z
2142:4
2138:N
2134:+
2129:2
2125:y
2119:4
2115:M
2111:+
2106:2
2102:x
2096:4
2092:L
2070:X
2049:.
2044:2
2040:)
2036:t
2031:2
2027:r
2023:+
2018:1
2014:r
2010:(
2007::
2002:2
1998:)
1994:t
1989:2
1985:q
1981:+
1976:1
1972:q
1968:(
1965::
1960:2
1956:)
1952:t
1947:2
1943:p
1939:+
1934:1
1930:p
1926:(
1923:=
1918:2
1914:X
1897:X
1889:t
1869:.
1866:)
1863:t
1858:2
1854:r
1850:+
1845:1
1841:r
1837:(
1834::
1831:)
1828:t
1823:2
1819:q
1815:+
1810:1
1806:q
1802:(
1799::
1796:)
1793:t
1788:2
1784:p
1780:+
1775:1
1771:p
1767:(
1764:=
1761:X
1731:2
1727:r
1723::
1718:2
1714:q
1710::
1705:2
1701:p
1674:1
1670:r
1666::
1661:1
1657:q
1653::
1648:1
1644:p
1628:.
1606:,
1603:0
1600:=
1597:b
1594:a
1591:w
1588:+
1585:a
1582:c
1579:v
1576:+
1573:c
1570:b
1567:u
1539:.
1509:=
1506:C
1497:w
1494:+
1491:B
1482:v
1479:+
1476:A
1467:u
1431:,
1428:0
1425:=
1422:b
1419:a
1416:v
1413:u
1410:2
1404:a
1401:c
1398:u
1395:w
1392:2
1386:c
1383:b
1380:w
1377:v
1374:2
1366:2
1362:c
1356:2
1352:w
1348:+
1343:2
1339:b
1333:2
1329:v
1325:+
1320:2
1316:a
1310:2
1306:u
1266:=
1263:z
1260:)
1257:p
1254:v
1251:+
1248:q
1245:u
1242:(
1239:+
1236:y
1233:)
1230:r
1227:u
1224:+
1221:p
1218:w
1215:(
1212:+
1209:x
1206:)
1203:q
1200:w
1197:+
1194:r
1191:v
1188:(
1172:P
1158:r
1155::
1152:q
1149::
1146:p
1143:=
1140:P
1114:)
1111:z
1108:a
1102:x
1099:c
1096:(
1093:)
1090:y
1087:c
1081:z
1078:b
1075:(
1072::
1069:)
1066:y
1063:c
1057:z
1054:b
1051:(
1048:)
1045:x
1042:b
1036:y
1033:a
1030:(
1027::
1024:)
1021:x
1018:b
1012:y
1009:a
1006:(
1003:)
1000:z
997:a
991:x
988:c
985:(
955:△
935:z
928:y
921:x
913:△
896:.
893:v
890:a
887:+
884:u
881:b
878::
875:u
872:c
869:+
866:w
863:a
860::
857:w
854:b
851:+
848:v
845:c
807:=
800:y
797:u
794:+
791:x
788:v
781:,
778:0
775:=
768:x
765:w
762:+
759:z
756:u
749:,
746:0
743:=
736:z
733:v
730:+
727:v
724:w
690:.
687:)
684:w
681:c
675:v
672:b
669:+
666:u
663:a
660:(
657:w
654::
651:)
648:w
645:c
642:+
639:v
636:b
630:u
627:a
624:(
621:v
618::
615:)
612:w
609:c
606:+
603:v
600:b
597:+
594:u
591:a
585:(
582:u
543:=
540:y
537:x
534:v
531:u
528:2
522:x
519:z
516:u
513:w
510:2
504:z
501:y
498:w
495:v
492:2
484:2
480:z
474:2
470:w
466:+
461:2
457:y
451:2
447:v
443:+
438:2
434:x
428:2
424:u
407:△
393:△
373:=
370:z
367:w
364:+
361:y
358:v
355:+
352:x
349:u
332:X
323:w
319:v
315:u
297:,
294:0
291:=
288:y
285:x
282:w
279:+
276:x
273:z
270:v
267:+
264:z
261:y
258:u
235:z
232::
229:y
226::
223:x
220:=
217:X
193:△
179:,
175:|
171:B
168:A
164:|
160:=
157:c
154:,
150:|
146:A
143:C
139:|
135:=
132:b
129:,
125:|
121:C
118:B
114:|
110:=
107:a
94:△
90:C
86:B
82:A
75:∠
71:A
60:△
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