2493:
2626:
2501:
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
2280:
835:
2070:
1452:
567:
2297:
1890:
1135:
1545:
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
2537:
711:
2302:
731:
1290:
1533:
200:
917:
1754:
1697:
1627:
318:
397:
1179:
256:
2097:
726:
1919:
2774:
1311:
429:
2488:{\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}}}
1767:
2654:
The lines connecting the tangency points of any inellipse of a triangle with the opposite vertices of the triangle are concurrent.
991:
2621:{\displaystyle {\frac {\text{Area of inellipse}}{\text{Area of triangle}}}=\pi {\sqrt {(1-2\alpha )(1-2\beta )(1-2\gamma )}},}
588:
1194:
1473:
2518:
17:
113:
851:
2698:, the unique ellipse that passes through a triangle's three vertices and is centered at the triangle's
1706:
1649:
1573:
264:
2871:
2517:. In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum
1635:
in which case it is tangent externally to one of the sides of the triangle and is tangent to the
355:
2713:
2717:
2695:
2275:{\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,}
1547:
1461:
1146:
980:
223:
213:
2707:, the unique conic which passes through a triangle's three vertices, its centroid, and its
8:
2731:
830:{\displaystyle {\begin{aligned}wv+vz&=0,\\uz+wx&=0,\\vx+uy&=0.\end{aligned}}}
409:
in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.
338:
31:
27:
Conic section that passes through the vertices of a triangle or is tangent to its sides
2760:
2754:
2721:
2510:
43:
2836:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.
2757:, the unique ellipse that is tangent to a triangle's three sides at their midpoints
55:
2503:
2849:
2796:
2763:, the unique ellipse tangent to a triangle's sides at the contact points of its
2769:
2704:
2065:{\displaystyle X^{2}=(p_{1}+p_{2}t)^{2}:(q_{1}+q_{2}t)^{2}:(r_{1}+r_{2}t)^{2}.}
2865:
2664:
1636:
1181:
is a point on the general circumconic, then the line tangent to the conic at
59:
39:
2855:
2808:
2738:, and various other notable points, and has center on the nine-point circle.
2725:
2685:
1903:
2795:
Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web
Resource.
2751:, the unique circle that is internally tangent to a triangle's three sides
2735:
2708:
2734:, a rectangular hyperbola that passes through a triangle's orthocenter,
2513:, also called the midpoint inellipse, with its center at the triangle's
2807:
Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web
Resource.
1447:{\displaystyle u^{2}a^{2}+v^{2}b^{2}+w^{2}c^{2}-2vwbc-2wuca-2uvab=0,}
562:{\displaystyle u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy=0.}
2764:
2748:
2699:
2668:
2514:
1561:
1542:
1299:
47:
2724:
and passing through the triangle's three vertices as well as its
2085:
2689:
1885:{\displaystyle X=(p_{1}+p_{2}t):(q_{1}+q_{2}t):(r_{1}+r_{2}t).}
922:
The lines tangent to the general inconic are the sidelines of
2634:
which is maximized by the centroid's barycentric coordinates
716:
The lines tangent to the general circumconic at the vertices
1130:{\displaystyle (cx-az)(ay-bx):(ay-bx)(bz-cy):(bz-cy)(cx-az)}
2832:
Chakerian, G. D. "A Distorted View of
Geometry." Ch. 7 in
2667:
fall on the line segment connecting the midpoints of the
2506:, the inellipse with its center at that point is unique.
964:
Each noncircular circumconic meets the circumcircle of
2540:
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706:{\displaystyle u(-au+bv+cw):v(au-bv+cw):w(au+bv-cw).}
591:
432:
358:
267:
226:
116:
1541:Of all triangles inscribed in a given ellipse, the
582:The center of the general circumconic is the point
2620:
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1527:
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1284:
1173:
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911:
829:
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561:
391:
312:
250:
194:
2863:
2728:, orthocenter, and various other notable centers
2692:that passes through a triangle's three vertices
845:The center of the general inconic is the point
84:denotes not only the vertex but also the angle
2658:
2797:http://mathworld.wolfram.com/Circumconic.html
1285:{\displaystyle (vr+wq)x+(wp+ur)y+(uq+vp)z=0.}
572:
2828:
2826:
2824:
2822:
2820:
2818:
2816:
2509:The inellipse with the largest area is the
69:are distinct non-collinear points, and let
1528:{\displaystyle u\cos A+v\cos B+w\cos C=0.}
2809:http://mathworld.wolfram.com/Inconic.html
2663:All the centers of inellipses of a given
2813:
76:denote the triangle whose vertices are
14:
2864:
1298:The general circumconic reduces to a
416:is tangent to the three sidelines of
195:{\displaystyle a=|BC|,b=|CA|,c=|AB|,}
402:This line meets the circumcircle of
24:
912:{\displaystyle cv+bw:aw+cu:bu+av.}
25:
2883:
2843:
1749:{\displaystyle p_{2}:q_{2}:r_{2}}
1692:{\displaystyle p_{1}:q_{1}:r_{1}}
1637:extensions of the other two sides
1560:The general inconic reduces to a
953:
220:is the locus of a variable point
958:
2084:is the inconic, necessarily an
345:on the circumconic, other than
2801:
2789:
2610:
2595:
2592:
2577:
2574:
2559:
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2020:
2008:
1978:
1966:
1936:
1876:
1847:
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1806:
1777:
1622:{\displaystyle ubc+vca+wab=0,}
1270:
1252:
1243:
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1106:
1103:
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1013:
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625:
595:
577:
313:{\displaystyle uyz+vzx+wxy=0,}
185:
174:
160:
149:
135:
124:
80:. Following common practice,
42:that passes through the three
13:
1:
2782:
2528:of the inellipse's center, is
423:and is given by the equation
1758:are distinct points, and let
1554:
977:fourth point of intersection
7:
2674:
2659:Extension to quadrilaterals
392:{\displaystyle ux+vy+wz=0.}
10:
2888:
840:
2720:centered on a triangle's
929:, given by the equations
573:Centers and tangent lines
349:, is a point on the line
2519:barycentric coordinates
2088:, given by the equation
1174:{\displaystyle P=p:q:r}
258:satisfying an equation
251:{\displaystyle X=x:y:z}
58:in the sides, possibly
2671:of the quadrilateral.
2622:
2489:
2276:
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1886:
1750:
1693:
1623:
1529:
1448:
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1175:
1131:
971:in a point other than
913:
831:
707:
563:
393:
314:
252:
196:
2718:rectangular hyperbola
2696:Steiner circumellipse
2623:
2490:
2277:
2067:
1887:
1751:
1694:
1624:
1548:Steiner circumellipse
1530:
1462:rectangular hyperbola
1449:
1287:
1176:
1132:
981:trilinear coordinates
914:
832:
708:
564:
394:
315:
253:
214:trilinear coordinates
197:
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1920:
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1195:
1147:
992:
852:
727:
589:
430:
356:
265:
224:
114:
95:, and similarly for
2732:Feuerbach hyperbola
1902:ranges through the
975:, often called the
720:are, respectively,
218:general circumconic
202:the sidelengths of
54:is a conic section
2834:Mathematical Plums
2618:
2485:
2483:
2272:
2062:
1910:is a line. Define
1882:
1746:
1689:
1619:
1525:
1444:
1282:
1171:
1127:
909:
827:
825:
703:
559:
389:
339:isogonal conjugate
310:
248:
192:
32:Euclidean geometry
2761:Mandart inellipse
2755:Steiner inellipse
2722:nine-point circle
2705:Kiepert hyperbola
2613:
2549:
2548:
2545:
2544:Area of inellipse
2511:Steiner inellipse
1898:As the parameter
62:, of a triangle.
16:(Redirected from
2879:
2837:
2830:
2811:
2805:
2799:
2793:
2770:Kiepert parabola
2648:
2627:
2625:
2624:
2619:
2614:
2558:
2550:
2547:Area of triangle
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2527:
2494:
2492:
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2477:
2476:
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2406:
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2279:
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2273:
2256:
2255:
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2223:
2214:
2213:
2192:
2191:
2182:
2181:
2166:
2165:
2156:
2155:
2143:
2142:
2133:
2132:
2120:
2119:
2110:
2109:
2083:
2071:
2069:
2068:
2063:
2058:
2057:
2045:
2044:
2032:
2031:
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2002:
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1989:
1974:
1973:
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1960:
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1947:
1932:
1931:
1909:
1901:
1891:
1889:
1888:
1883:
1872:
1871:
1859:
1858:
1837:
1836:
1824:
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1802:
1801:
1789:
1788:
1757:
1755:
1753:
1752:
1747:
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1732:
1731:
1719:
1718:
1700:
1698:
1696:
1695:
1690:
1688:
1687:
1675:
1674:
1662:
1661:
1628:
1626:
1625:
1620:
1534:
1532:
1531:
1526:
1453:
1451:
1450:
1445:
1380:
1379:
1370:
1369:
1357:
1356:
1347:
1346:
1334:
1333:
1324:
1323:
1291:
1289:
1288:
1283:
1184:
1180:
1178:
1177:
1172:
1136:
1134:
1133:
1128:
974:
970:
949:
942:
935:
928:
918:
916:
915:
910:
836:
834:
833:
828:
826:
719:
712:
710:
709:
704:
568:
566:
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560:
498:
497:
488:
487:
475:
474:
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464:
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451:
442:
441:
422:
408:
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390:
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21:
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2790:
2785:
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2661:
2635:
2557:
2541:
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2504:medial triangle
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2422:
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2079:
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2049:
2040:
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2027:
2023:
2011:
2007:
1998:
1994:
1985:
1981:
1969:
1965:
1956:
1952:
1943:
1939:
1927:
1923:
1921:
1918:
1917:
1907:
1906:, the locus of
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1375:
1371:
1365:
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1325:
1319:
1315:
1313:
1310:
1309:
1196:
1193:
1192:
1182:
1148:
1145:
1144:
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990:
989:
972:
965:
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956:
944:
937:
930:
923:
853:
850:
849:
843:
824:
823:
813:
795:
794:
781:
763:
762:
749:
730:
728:
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724:
717:
590:
587:
586:
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575:
493:
489:
483:
479:
470:
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460:
456:
447:
443:
437:
433:
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428:
427:
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414:general inconic
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324:
323:for some point
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2872:Conic sections
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2844:External links
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1000:
997:
984:
983:
960:
957:
955:
954:Other features
952:
920:
919:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
842:
839:
838:
837:
822:
819:
816:
814:
812:
809:
806:
803:
800:
797:
796:
793:
790:
787:
784:
782:
780:
777:
774:
771:
768:
765:
764:
761:
758:
755:
752:
750:
748:
745:
742:
739:
736:
733:
732:
714:
713:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
579:
576:
574:
571:
570:
569:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
496:
492:
486:
482:
478:
473:
469:
463:
459:
455:
450:
446:
440:
436:
400:
399:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
341:of each point
321:
320:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
247:
244:
241:
238:
235:
232:
229:
191:
187:
183:
180:
176:
172:
169:
166:
162:
158:
155:
151:
147:
144:
141:
137:
133:
130:
126:
122:
119:
26:
9:
6:
4:
3:
2:
2884:
2873:
2870:
2869:
2867:
2857:
2854:
2851:
2848:
2847:
2835:
2829:
2827:
2825:
2823:
2821:
2819:
2817:
2810:
2804:
2798:
2792:
2788:
2776:
2773:
2771:
2768:
2766:
2762:
2759:
2756:
2753:
2750:
2747:
2746:
2745:
2742:
2737:
2733:
2730:
2727:
2723:
2719:
2716:hyperbola, a
2715:
2712:
2710:
2706:
2703:
2701:
2697:
2694:
2691:
2688:, the unique
2687:
2684:
2683:
2682:
2679:
2678:
2672:
2670:
2666:
2665:quadrilateral
2653:
2652:
2646:
2642:
2638:
2633:
2632:
2615:
2607:
2604:
2601:
2598:
2589:
2586:
2583:
2580:
2571:
2568:
2565:
2562:
2554:
2551:
2534:
2533:
2532:
2531:
2525:
2520:
2516:
2512:
2508:
2505:
2500:
2499:
2478:
2473:
2469:
2463:
2459:
2455:
2450:
2446:
2440:
2436:
2432:
2430:
2425:
2418:
2413:
2409:
2403:
2399:
2395:
2390:
2386:
2380:
2376:
2372:
2370:
2365:
2358:
2353:
2349:
2343:
2339:
2335:
2330:
2326:
2320:
2316:
2312:
2310:
2305:
2294:
2293:
2292:
2291:
2287:
2286:
2269:
2266:
2263:
2260:
2257:
2252:
2248:
2242:
2238:
2234:
2231:
2228:
2225:
2220:
2216:
2210:
2206:
2202:
2199:
2196:
2193:
2188:
2184:
2178:
2174:
2170:
2167:
2162:
2158:
2152:
2148:
2144:
2139:
2135:
2129:
2125:
2121:
2116:
2112:
2106:
2102:
2094:
2093:
2092:
2091:
2087:
2082:
2078:The locus of
2077:
2076:
2059:
2054:
2046:
2041:
2037:
2033:
2028:
2024:
2017:
2012:
2004:
1999:
1995:
1991:
1986:
1982:
1975:
1970:
1962:
1957:
1953:
1949:
1944:
1940:
1933:
1928:
1924:
1916:
1915:
1914:
1913:
1905:
1897:
1896:
1879:
1873:
1868:
1864:
1860:
1855:
1851:
1844:
1838:
1833:
1829:
1825:
1820:
1816:
1809:
1803:
1798:
1794:
1790:
1785:
1781:
1774:
1771:
1764:
1763:
1762:
1761:
1741:
1737:
1733:
1728:
1724:
1720:
1715:
1711:
1684:
1680:
1676:
1671:
1667:
1663:
1658:
1654:
1644:Suppose that
1643:
1642:
1638:
1634:
1633:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1570:
1569:
1568:
1567:
1563:
1559:
1558:
1549:
1544:
1540:
1539:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1470:
1469:
1468:
1467:
1463:
1459:
1458:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1376:
1372:
1366:
1362:
1358:
1353:
1349:
1343:
1339:
1335:
1330:
1326:
1320:
1316:
1308:
1307:
1306:
1305:
1301:
1297:
1296:
1279:
1276:
1273:
1267:
1264:
1261:
1258:
1255:
1249:
1246:
1240:
1237:
1234:
1231:
1228:
1222:
1219:
1213:
1210:
1207:
1204:
1201:
1191:
1190:
1189:
1188:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1142:
1141:
1121:
1118:
1115:
1112:
1109:
1100:
1097:
1094:
1091:
1088:
1082:
1076:
1073:
1070:
1067:
1064:
1055:
1052:
1049:
1046:
1043:
1037:
1031:
1028:
1025:
1022:
1019:
1010:
1007:
1004:
1001:
998:
988:
987:
986:
985:
982:
978:
969:
963:
962:
951:
947:
940:
933:
927:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
848:
847:
846:
820:
817:
815:
810:
807:
804:
801:
798:
791:
788:
785:
783:
778:
775:
772:
769:
766:
759:
756:
753:
751:
746:
743:
740:
737:
734:
723:
722:
721:
700:
694:
691:
688:
685:
682:
679:
676:
673:
667:
664:
658:
655:
652:
649:
646:
643:
640:
637:
631:
628:
622:
619:
616:
613:
610:
607:
604:
601:
598:
592:
585:
584:
583:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
494:
490:
484:
480:
476:
471:
467:
461:
457:
453:
448:
444:
438:
434:
426:
425:
424:
421:
415:
410:
407:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
352:
351:
350:
340:
335:
331:
327:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
261:
260:
259:
245:
242:
239:
236:
233:
230:
227:
219:
215:
210:
207:
189:
181:
178:
170:
167:
164:
156:
153:
145:
142:
139:
131:
128:
120:
117:
108:
103:as angles in
89:
74:
63:
61:
57:
53:
49:
45:
41:
40:conic section
37:
33:
19:
18:Circumellipse
2858:at MathWorld
2852:at MathWorld
2833:
2803:
2791:
2775:Yff parabola
2743:
2726:circumcenter
2686:Circumcircle
2681:Circumconics
2680:
2662:
2644:
2640:
2636:
2523:
2080:
1904:real numbers
976:
967:
945:
938:
931:
925:
921:
844:
715:
581:
419:
413:
411:
405:
401:
333:
329:
325:
322:
217:
211:
205:
106:
87:
72:
64:
51:
35:
29:
2850:Circumconic
2736:Nagel point
2709:orthocenter
1185:is given by
979:, given by
959:Circumconic
578:Circumconic
36:circumconic
2783:References
91:at vertex
2765:excircles
2669:diagonals
2608:γ
2602:−
2590:β
2584:−
2572:α
2566:−
2555:π
2456:−
2396:−
2336:−
2232:−
2200:−
2168:−
1514:
1499:
1484:
1460:and to a
1418:−
1400:−
1382:−
1116:−
1095:−
1071:−
1050:−
1026:−
1005:−
689:−
644:−
599:−
536:−
518:−
500:−
56:inscribed
50:, and an
2866:Category
2749:Incircle
2744:Inconics
2700:centroid
2675:Examples
2515:centroid
1562:parabola
1543:centroid
1300:parabola
332: :
328: :
65:Suppose
60:extended
48:triangle
44:vertices
2856:Inconic
2714:Jeřábek
2524:α, β, γ
2086:ellipse
1756:
1703:
1699:
1646:
1555:Inconic
973:A, B, C
841:Inconic
718:A, B, C
347:A, B, C
337:. The
110:. Let
78:A, B, C
67:A, B, C
52:inconic
2690:circle
216:, the
2288:where
46:of a
38:is a
1701:and
412:The
99:and
34:, a
2647:= ⅓
1511:cos
1496:cos
1481:cos
1143:If
968:ABC
948:= 0
941:= 0
934:= 0
926:ABC
420:ABC
406:ABC
212:In
206:ABC
107:ABC
88:BAC
73:ABC
30:In
2868::
2815:^
2643:=
2639:=
1523:0.
1280:0.
950:.
943:,
936:,
821:0.
557:0.
387:0.
209:.
2649:.
2645:γ
2641:β
2637:α
2616:,
2611:)
2605:2
2599:1
2596:(
2593:)
2587:2
2581:1
2578:(
2575:)
2569:2
2563:1
2560:(
2552:=
2526:)
2522:(
2479:.
2474:2
2470:p
2464:1
2460:q
2451:2
2447:q
2441:1
2437:p
2433:=
2426:N
2419:,
2414:2
2410:r
2404:1
2400:p
2391:2
2387:p
2381:1
2377:r
2373:=
2366:M
2359:,
2354:2
2350:q
2344:1
2340:r
2331:2
2327:r
2321:1
2317:q
2313:=
2306:L
2270:,
2267:0
2264:=
2261:y
2258:x
2253:2
2249:M
2243:2
2239:L
2235:2
2229:x
2226:z
2221:2
2217:L
2211:2
2207:N
2203:2
2197:z
2194:y
2189:2
2185:N
2179:2
2175:M
2171:2
2163:2
2159:z
2153:4
2149:N
2145:+
2140:2
2136:y
2130:4
2126:M
2122:+
2117:2
2113:x
2107:4
2103:L
2081:X
2060:.
2055:2
2051:)
2047:t
2042:2
2038:r
2034:+
2029:1
2025:r
2021:(
2018::
2013:2
2009:)
2005:t
2000:2
1996:q
1992:+
1987:1
1983:q
1979:(
1976::
1971:2
1967:)
1963:t
1958:2
1954:p
1950:+
1945:1
1941:p
1937:(
1934:=
1929:2
1925:X
1908:X
1900:t
1880:.
1877:)
1874:t
1869:2
1865:r
1861:+
1856:1
1852:r
1848:(
1845::
1842:)
1839:t
1834:2
1830:q
1826:+
1821:1
1817:q
1813:(
1810::
1807:)
1804:t
1799:2
1795:p
1791:+
1786:1
1782:p
1778:(
1775:=
1772:X
1742:2
1738:r
1734::
1729:2
1725:q
1721::
1716:2
1712:p
1685:1
1681:r
1677::
1672:1
1668:q
1664::
1659:1
1655:p
1639:.
1617:,
1614:0
1611:=
1608:b
1605:a
1602:w
1599:+
1596:a
1593:c
1590:v
1587:+
1584:c
1581:b
1578:u
1550:.
1520:=
1517:C
1508:w
1505:+
1502:B
1493:v
1490:+
1487:A
1478:u
1442:,
1439:0
1436:=
1433:b
1430:a
1427:v
1424:u
1421:2
1415:a
1412:c
1409:u
1406:w
1403:2
1397:c
1394:b
1391:w
1388:v
1385:2
1377:2
1373:c
1367:2
1363:w
1359:+
1354:2
1350:b
1344:2
1340:v
1336:+
1331:2
1327:a
1321:2
1317:u
1277:=
1274:z
1271:)
1268:p
1265:v
1262:+
1259:q
1256:u
1253:(
1250:+
1247:y
1244:)
1241:r
1238:u
1235:+
1232:p
1229:w
1226:(
1223:+
1220:x
1217:)
1214:q
1211:w
1208:+
1205:r
1202:v
1199:(
1183:P
1169:r
1166::
1163:q
1160::
1157:p
1154:=
1151:P
1125:)
1122:z
1119:a
1113:x
1110:c
1107:(
1104:)
1101:y
1098:c
1092:z
1089:b
1086:(
1083::
1080:)
1077:y
1074:c
1068:z
1065:b
1062:(
1059:)
1056:x
1053:b
1047:y
1044:a
1041:(
1038::
1035:)
1032:x
1029:b
1023:y
1020:a
1017:(
1014:)
1011:z
1008:a
1002:x
999:c
996:(
966:△
946:z
939:y
932:x
924:△
907:.
904:v
901:a
898:+
895:u
892:b
889::
886:u
883:c
880:+
877:w
874:a
871::
868:w
865:b
862:+
859:v
856:c
818:=
811:y
808:u
805:+
802:x
799:v
792:,
789:0
786:=
779:x
776:w
773:+
770:z
767:u
760:,
757:0
754:=
747:z
744:v
741:+
738:v
735:w
701:.
698:)
695:w
692:c
686:v
683:b
680:+
677:u
674:a
671:(
668:w
665::
662:)
659:w
656:c
653:+
650:v
647:b
641:u
638:a
635:(
632:v
629::
626:)
623:w
620:c
617:+
614:v
611:b
608:+
605:u
602:a
596:(
593:u
554:=
551:y
548:x
545:v
542:u
539:2
533:x
530:z
527:u
524:w
521:2
515:z
512:y
509:w
506:v
503:2
495:2
491:z
485:2
481:w
477:+
472:2
468:y
462:2
458:v
454:+
449:2
445:x
439:2
435:u
418:△
404:△
384:=
381:z
378:w
375:+
372:y
369:v
366:+
363:x
360:u
343:X
334:w
330:v
326:u
308:,
305:0
302:=
299:y
296:x
293:w
290:+
287:x
284:z
281:v
278:+
275:z
272:y
269:u
246:z
243::
240:y
237::
234:x
231:=
228:X
204:△
190:,
186:|
182:B
179:A
175:|
171:=
168:c
165:,
161:|
157:A
154:C
150:|
146:=
143:b
140:,
136:|
132:C
129:B
125:|
121:=
118:a
105:△
101:C
97:B
93:A
86:∠
82:A
71:△
20:)
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