425:
388:
20:
2888:
1666:
982:
2642:
1854:
1454:
735:
The proofs of properties a),b),c) are based on the following properties of an affine mapping: 1) any triangle can be considered as an affine image of an equilateral triangle. 2) Midpoints of sides are mapped onto midpoints and centroids on centroids. The center of an ellipse is mapped onto the center
790:
232:
2883:{\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.}
1297:
1432:
1694:
1661:{\displaystyle {\begin{aligned}M&:={\color {blue}{\frac {1}{4}}}\left({\vec {SC}}^{2}+{\frac {1}{3}}{\vec {AB}}^{2}\right)\\N&:={\frac {1}{{\color {blue}4}{\sqrt {3}}}}\left|\det \left({\vec {SC}},{\vec {AB}}\right)\right|\end{aligned}}}
1129:
2906:-gons have an interior ellipse that is tangent to each side at the side's midpoint. Marden's theorem still applies: the foci of the Steiner inellipse are zeroes of the derivative of the polynomial whose zeroes are the vertices of the
977:{\displaystyle {\vec {x}}={\vec {p}}(t)={\overrightarrow {OS}}\;+\;{\color {blue}{\frac {1}{2}}}{\overrightarrow {SC}}\;\cos t\;+\;{\frac {1}{{\color {blue}2}{\sqrt {3}}}}{\overrightarrow {AB}}\;\sin t\;,\quad 0\leq t<2\pi \ .}
1970:
2388:
46:
2134:
2247:
1699:
1459:
1149:
51:
1302:
719:
662:
598:
2618:
The foci of the
Steiner inellipse of a triangle are the intersections of the inellipse's major axis and the circle with center on the minor axis and going through the Fermat points.
2602:
2530:
1849:{\displaystyle {\begin{aligned}a&={\frac {1}{2}}\left({\sqrt {M+2N}}+{\sqrt {M-2N}}\right)\\b&={\frac {1}{2}}\left({\sqrt {M+2N}}-{\sqrt {M-2N}}\right)\ .\end{aligned}}}
520:
371:
744:. It touches the sides at its midpoints. There is no other (non-degenerate) conic section with the same properties, because a conic section is determined by 5 points/tangents.
2480:
1003:
2607:
The axes of the
Steiner inellipse of a triangle are tangent to its Kiepert parabola, the unique parabola that is tangent to the sides of the triangle and has the
296:, also called simply the Steiner ellipse, which is the unique ellipse that passes through the vertices of a given triangle and whose center is the triangle's
1878:
777:
is a scaled
Steiner ellipse (factor 1/2, center is centroid) one gets a parametric representation derived from the trigonometric representation of the
2258:
227:{\displaystyle {\begin{aligned}&D_{x}(1+7i-x)(7+5i-x)(3+i-x)\\&=-3\left({\tfrac {13}{3}}+{\tfrac {11}{3}}i-x\right)(3+5i-x)\end{aligned}}}
2962:
748:
c) The circumcircle is mapped by a scaling, with factor 1/2 and the centroid as center, onto the incircle. The eccentricity is an invariant.
1996:
3050:, The Dolciani Mathematical Expositions, vol. 4, Washington, D.C.: Mathematical Association of America, pp. 135–136, 145–146
2173:
3168:
1292:{\displaystyle \cot(2t_{0})={\tfrac {{\vec {f}}_{1}^{\,2}-{\vec {f}}_{2}^{\,2}}{2{\vec {f}}_{1}\cdot {\vec {f}}_{2}}}\quad }
1427:{\displaystyle \quad {\vec {f}}_{1}={\frac {1}{2}}{\vec {SC}},\quad {\vec {f}}_{2}={\frac {1}{2{\sqrt {3}}}}{\vec {AB}}\ .}
741:
750:
d) The ratio of areas is invariant to affine transformations. So the ratio can be calculated for the equilateral triangle.
778:
686:
3069:
614:
281:
of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is
1673:
553:
3077:
2970:
2537:
2489:
3131:
Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity",
670:
472:
323:
753:
669:
Steiner
Ellipse with scaling factor 1/2 and the centroid as center. Hence both ellipses have the same
2612:
2405:
1124:{\displaystyle {\vec {p}}(t_{0}),\;{\vec {p}}(t_{0}\pm {\frac {\pi }{2}}),\;{\vec {p}}(t_{0}+\pi ),}
3178:
2439:
3173:
3003:
1983:
293:
3133:
3098:
2999:
433:
282:
3025:
2484:
respectively. The major axis of the triangle's
Steiner inellipse is the inner bisector of
8:
2604:
that is, the sum and difference of the distances of the Fermat points from the centroid.
2394:
24:
2143:
is an arbitrary positive constant times the distance of a point from the side of length
3116:
2987:
3146:
3117:
Scimemi, Benedetto, "Simple
Relations Regarding the Steiner Inellipse of a Triangle",
3022:
2932:
2424:
2398:
278:
3046:
Chakerian, G. D. (1979), "A distorted view of geometry", in
Honsberger, Ross (ed.),
3090:
3086:
2983:
2979:
2413:
274:
32:
3094:
2995:
1965:{\displaystyle c={\sqrt {a^{2}-b^{2}}}=\dotsb ={\sqrt {\sqrt {M^{2}-4N^{2}}}}\ .}
2402:
2383:{\displaystyle Z={\sqrt {a^{4}+b^{4}+c^{4}-a^{2}b^{2}-b^{2}c^{2}-c^{2}a^{2}}}.}
3162:
286:
2931:
Weisstein, E. "Steiner
Inellipse" — From MathWorld, A Wolfram Web Resource,
2163:
The lengths of the semi-major and semi-minor axes for a triangle with sides
2431:
738:
Hence its suffice to prove properties a),b),c) for an equilateral triangle:
3065:
2991:
2946:
100 Great
Problems of Elementary Mathematics, Their History and Solution
2608:
2417:
2409:
1990:(with these parameters having a different meaning than previously) is
424:
387:
3030:
270:
16:
Unique ellipse tangent to all 3 midpoints of a given triangle's sides
2129:{\displaystyle a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}-2abxy-2bcyz-2cazx=0}
19:
535:
297:
266:
254:
238:
262:
258:
2242:{\displaystyle {\frac {1}{6}}{\sqrt {a^{2}+b^{2}+c^{2}\pm 2Z}},}
3147:
Parish, James L., "On the derivative of a vertex polynomial",
3020:
2898:
The
Steiner inellipse of a triangle can be generalized to
759:
289:, and a proof of its uniqueness is given by Dan Kalman.
1182:
691:
313:
An ellipse that is tangent to the sides of a triangle
171:
156:
2948:(trans. D. Antin), Dover, New York, 1965, problem 98.
2645:
2540:
2492:
2442:
2261:
2176:
1999:
1881:
1697:
1457:
1305:
1152:
1006:
793:
689:
617:
556:
475:
326:
49:
285:. The Steiner inellipse is attributed by Dörrie to
2933:http://mathworld.wolfram.com/SteinerInellipse.html
2882:
2596:
2524:
2474:
2382:
2241:
2128:
1964:
1848:
1660:
1426:
1291:
1123:
976:
713:
656:
592:
514:
365:
226:
2590:
3160:
1599:
2423:The major axis of the Steiner inellipse is the
740:a) To any equilateral triangle there exists an
665:c2) The Steiner inellipse of a triangle is the
522:of its sides the following statements are true:
2416:of the Steiner inellipse are the zeros of the
2430:Denote the centroid and the first and second
714:{\displaystyle {\tfrac {\pi }{3{\sqrt {3}}}}}
3070:"Triangles, ellipses, and cubic polynomials"
2621:As with any ellipse inscribed in a triangle
303:
3064:
2927:
2925:
2923:
1083:
1038:
945:
935:
896:
892:
882:
852:
848:
770:Because a Steiner inellipse of a triangle
657:{\displaystyle \triangle M_{1}M_{2}M_{3}.}
3112:
3110:
3108:
3045:
2963:"An elementary proof of Marden's theorem"
1982:The equation of the Steiner inellipse in
1232:
1204:
593:{\displaystyle \triangle M_{1}M_{2}M_{3}}
292:The Steiner inellipse contrasts with the
3039:
2956:
2954:
2920:
423:
386:
18:
3060:
3058:
2597:{\displaystyle |GF_{-}|\pm |GF_{+}|\!;}
2155:with the same multiplicative constant.
760:Parametric representation and semi-axes
611:is the Steiner ellipse of the triangle
3161:
3105:
2960:
3021:
2951:
1977:
904:
854:
3151:6, 2006, pp. 285–288: Proposition 5.
3055:
23:The Steiner Inellipse. According to
2525:{\displaystyle \angle F_{+}GF_{-}.}
2158:
727:of all inellipses of the triangle.
27:, given the triangle with vertices
13:
2493:
1578:
1473:
618:
557:
14:
3190:
2893:
1986:for a triangle with side lengths
723:e) The Steiner inellipse has the
515:{\displaystyle M_{1},M_{2},M_{3}}
366:{\displaystyle M_{1},M_{2},M_{3}}
721:-times the area of the triangle.
534:of the Steiner inellipse is the
1359:
1306:
1288:
949:
3140:
3125:
3091:10.1080/00029890.2008.11920581
3014:
2984:10.1080/00029890.2008.11920532
2938:
2586:
2568:
2560:
2542:
1638:
1618:
1540:
1503:
1412:
1367:
1350:
1314:
1272:
1250:
1220:
1192:
1175:
1159:
1115:
1096:
1090:
1077:
1051:
1045:
1032:
1019:
1013:
827:
821:
815:
800:
528:exactly one Steiner inellipse.
261:inscribed in the triangle and
217:
196:
131:
113:
110:
89:
86:
65:
1:
3169:Curves defined for a triangle
3078:American Mathematical Monthly
2971:American Mathematical Monthly
2913:
2475:{\displaystyle G,F_{+},F_{-}}
604:and the Steiner inellipse of
2866:
2848:
2831:
2813:
2789:
2771:
2754:
2736:
2712:
2694:
2677:
2659:
2534:The lengths of the axes are
994:of the Steiner inellipse are
683:of the Steiner inellipse is
7:
2425:line of best orthogonal fit
1869:of the Steiner inellipse is
746:b) By a simple calculation.
10:
3195:
765:Parametric representation:
462:For an arbitrary triangle
419: Major and minor axes
395: Arbitrary triangle
304:Definition and properties
269:. It is an example of an
3137:96, March 2012, 161-165.
2401:of the triangle are the
3026:"Steiner Circumellipse"
447: Steiner inellipse
407: Steiner inellipse
2884:
2628:, letting the foci be
2598:
2526:
2476:
2384:
2243:
2130:
1966:
1850:
1662:
1445:With the abbreviations
1428:
1293:
1125:
978:
715:
658:
600:has the same centroid
594:
516:
455:
421:
367:
265:to the sides at their
234:
228:
29:(1, 7), (7, 5), (3, 1)
3068:; Phelps, S. (2008),
2885:
2599:
2527:
2477:
2385:
2244:
2131:
1984:trilinear coordinates
1967:
1851:
1663:
1429:
1294:
1126:
979:
716:
659:
595:
517:
453: Steiner ellipse
427:
413: Steiner ellipse
390:
368:
294:Steiner circumellipse
229:
35:of the inellipse are
22:
3134:Mathematical Gazette
2961:Kalman, Dan (2008),
2643:
2538:
2490:
2440:
2259:
2174:
2147:, and similarly for
1997:
1879:
1695:
1455:
1303:
1150:
1004:
791:
687:
615:
554:
473:
434:Equilateral triangle
324:
273:. By comparison the
47:
3149:Forum Geometricorum
3119:Forum Geometricorum
2420:of the polynomial.
1864:linear eccentricity
1237:
1209:
1144:is the solution of
3048:Mathematical plums
3023:Weisstein, Eric W.
2880:
2594:
2522:
2472:
2427:for the vertices.
2380:
2239:
2126:
1978:Trilinear equation
1962:
1846:
1844:
1658:
1656:
1582:
1484:
1424:
1289:
1286:
1213:
1185:
1121:
974:
908:
865:
711:
709:
654:
590:
512:
456:
422:
363:
247:midpoint inellipse
235:
224:
222:
180:
165:
2872:
2869:
2851:
2834:
2816:
2795:
2792:
2774:
2757:
2739:
2718:
2715:
2697:
2680:
2662:
2434:of a triangle as
2375:
2234:
2185:
1958:
1954:
1953:
1913:
1838:
1829:
1810:
1789:
1760:
1741:
1720:
1672:one gets for the
1641:
1621:
1592:
1589:
1543:
1526:
1506:
1482:
1420:
1415:
1399:
1396:
1370:
1353:
1337:
1317:
1285:
1275:
1253:
1223:
1195:
1093:
1075:
1048:
1016:
970:
933:
918:
915:
880:
863:
846:
818:
803:
708:
705:
550:c1) The triangle
375:Steiner inellipse
320:at its midpoints
279:Mandart inellipse
243:Steiner inellipse
179:
164:
3186:
3153:
3144:
3138:
3129:
3123:
3121:10, 2010: 55–77.
3114:
3103:
3101:
3074:
3062:
3053:
3051:
3043:
3037:
3036:
3035:
3018:
3012:
3010:
3008:
3002:, archived from
2967:
2958:
2949:
2942:
2936:
2929:
2909:
2905:
2901:
2889:
2887:
2886:
2881:
2873:
2871:
2870:
2865:
2857:
2852:
2847:
2839:
2836:
2835:
2830:
2822:
2817:
2812:
2804:
2801:
2796:
2794:
2793:
2788:
2780:
2775:
2770:
2762:
2759:
2758:
2753:
2745:
2740:
2735:
2727:
2724:
2719:
2717:
2716:
2711:
2703:
2698:
2693:
2685:
2682:
2681:
2676:
2668:
2663:
2658:
2650:
2647:
2635:
2631:
2627:
2603:
2601:
2600:
2595:
2589:
2584:
2583:
2571:
2563:
2558:
2557:
2545:
2533:
2531:
2529:
2528:
2523:
2518:
2517:
2505:
2504:
2483:
2481:
2479:
2478:
2473:
2471:
2470:
2458:
2457:
2395:Marden's theorem
2389:
2387:
2386:
2381:
2376:
2374:
2373:
2364:
2363:
2351:
2350:
2341:
2340:
2328:
2327:
2318:
2317:
2305:
2304:
2292:
2291:
2279:
2278:
2269:
2248:
2246:
2245:
2240:
2235:
2224:
2223:
2211:
2210:
2198:
2197:
2188:
2186:
2178:
2166:
2159:Other properties
2154:
2150:
2146:
2142:
2135:
2133:
2132:
2127:
2065:
2064:
2055:
2054:
2042:
2041:
2032:
2031:
2019:
2018:
2009:
2008:
1989:
1971:
1969:
1968:
1963:
1956:
1955:
1952:
1951:
1936:
1935:
1926:
1925:
1914:
1912:
1911:
1899:
1898:
1889:
1868:
1855:
1853:
1852:
1847:
1845:
1836:
1835:
1831:
1830:
1816:
1811:
1797:
1790:
1782:
1766:
1762:
1761:
1747:
1742:
1728:
1721:
1713:
1688:
1678:
1667:
1665:
1664:
1659:
1657:
1653:
1649:
1648:
1644:
1643:
1642:
1637:
1629:
1623:
1622:
1617:
1609:
1593:
1591:
1590:
1585:
1583:
1572:
1556:
1552:
1551:
1550:
1545:
1544:
1539:
1531:
1527:
1519:
1514:
1513:
1508:
1507:
1502:
1494:
1485:
1483:
1475:
1433:
1431:
1430:
1425:
1418:
1417:
1416:
1411:
1403:
1400:
1398:
1397:
1392:
1383:
1378:
1377:
1372:
1371:
1363:
1355:
1354:
1349:
1341:
1338:
1330:
1325:
1324:
1319:
1318:
1310:
1298:
1296:
1295:
1290:
1287:
1284:
1283:
1282:
1277:
1276:
1268:
1261:
1260:
1255:
1254:
1246:
1238:
1236:
1230:
1225:
1224:
1216:
1208:
1202:
1197:
1196:
1188:
1183:
1174:
1173:
1143:
1130:
1128:
1127:
1122:
1108:
1107:
1095:
1094:
1086:
1076:
1068:
1063:
1062:
1050:
1049:
1041:
1031:
1030:
1018:
1017:
1009:
983:
981:
980:
975:
968:
934:
929:
921:
919:
917:
916:
911:
909:
898:
881:
876:
868:
866:
864:
856:
847:
842:
834:
820:
819:
811:
805:
804:
796:
776:
720:
718:
717:
712:
710:
707:
706:
701:
692:
663:
661:
660:
655:
650:
649:
640:
639:
630:
629:
610:
603:
599:
597:
596:
591:
589:
588:
579:
578:
569:
568:
547:
540:
521:
519:
518:
513:
511:
510:
498:
497:
485:
484:
468:
452:
446:
441:
431:
418:
412:
406:
401:
394:
383:
372:
370:
369:
364:
362:
361:
349:
348:
336:
335:
319:
275:inscribed circle
251:midpoint ellipse
233:
231:
230:
225:
223:
195:
191:
181:
172:
166:
157:
137:
64:
63:
53:
42:
38:
30:
25:Marden's theorem
3194:
3193:
3189:
3188:
3187:
3185:
3184:
3183:
3179:Affine geometry
3159:
3158:
3157:
3156:
3145:
3141:
3130:
3126:
3115:
3106:
3072:
3063:
3056:
3044:
3040:
3019:
3015:
3006:
2965:
2959:
2952:
2943:
2939:
2930:
2921:
2916:
2907:
2903:
2899:
2896:
2858:
2856:
2840:
2838:
2837:
2823:
2821:
2805:
2803:
2802:
2800:
2781:
2779:
2763:
2761:
2760:
2746:
2744:
2728:
2726:
2725:
2723:
2704:
2702:
2686:
2684:
2683:
2669:
2667:
2651:
2649:
2648:
2646:
2644:
2641:
2640:
2633:
2629:
2622:
2585:
2579:
2575:
2567:
2559:
2553:
2549:
2541:
2539:
2536:
2535:
2513:
2509:
2500:
2496:
2491:
2488:
2487:
2485:
2466:
2462:
2453:
2449:
2441:
2438:
2437:
2435:
2397:, if the three
2369:
2365:
2359:
2355:
2346:
2342:
2336:
2332:
2323:
2319:
2313:
2309:
2300:
2296:
2287:
2283:
2274:
2270:
2268:
2260:
2257:
2256:
2219:
2215:
2206:
2202:
2193:
2189:
2187:
2177:
2175:
2172:
2171:
2164:
2161:
2152:
2148:
2144:
2140:
2060:
2056:
2050:
2046:
2037:
2033:
2027:
2023:
2014:
2010:
2004:
2000:
1998:
1995:
1994:
1987:
1980:
1947:
1943:
1931:
1927:
1924:
1907:
1903:
1894:
1890:
1888:
1880:
1877:
1876:
1866:
1843:
1842:
1815:
1796:
1795:
1791:
1781:
1774:
1768:
1767:
1746:
1727:
1726:
1722:
1712:
1705:
1698:
1696:
1693:
1692:
1680:
1676:
1655:
1654:
1630:
1628:
1627:
1610:
1608:
1607:
1606:
1602:
1598:
1594:
1584:
1577:
1576:
1571:
1564:
1558:
1557:
1546:
1532:
1530:
1529:
1528:
1518:
1509:
1495:
1493:
1492:
1491:
1490:
1486:
1474:
1472:
1465:
1458:
1456:
1453:
1452:
1404:
1402:
1401:
1391:
1387:
1382:
1373:
1362:
1361:
1360:
1342:
1340:
1339:
1329:
1320:
1309:
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1300:
1278:
1267:
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1187:
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1142:
1136:
1103:
1099:
1085:
1084:
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1040:
1039:
1026:
1022:
1008:
1007:
1005:
1002:
1001:
922:
920:
910:
903:
902:
897:
869:
867:
855:
853:
835:
833:
810:
809:
795:
794:
792:
789:
788:
779:Steiner ellipse
771:
762:
751:
749:
747:
745:
739:
737:
722:
700:
696:
690:
688:
685:
684:
678:
664:
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584:
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560:
555:
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551:
549:
542:
538:
529:
523:
506:
502:
493:
489:
480:
476:
474:
471:
470:
469:with midpoints
463:
461:
454:
450:
448:
444:
442:
436:
429:
420:
416:
414:
410:
408:
404:
402:
396:
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378:
357:
353:
344:
340:
331:
327:
325:
322:
321:
314:
306:
221:
220:
170:
155:
154:
150:
135:
134:
59:
55:
50:
48:
45:
44:
40:
36:
28:
17:
12:
11:
5:
3192:
3182:
3181:
3176:
3174:Conic sections
3171:
3155:
3154:
3139:
3124:
3104:
3085:(8): 679–689,
3054:
3038:
3013:
2978:(4): 330–338,
2950:
2937:
2918:
2917:
2915:
2912:
2895:
2894:Generalization
2892:
2891:
2890:
2879:
2876:
2868:
2864:
2861:
2855:
2850:
2846:
2843:
2833:
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2815:
2811:
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2799:
2791:
2787:
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2773:
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2756:
2752:
2749:
2743:
2738:
2734:
2731:
2722:
2714:
2710:
2707:
2701:
2696:
2692:
2689:
2679:
2675:
2672:
2666:
2661:
2657:
2654:
2593:
2588:
2582:
2578:
2574:
2570:
2566:
2562:
2556:
2552:
2548:
2544:
2521:
2516:
2512:
2508:
2503:
2499:
2495:
2469:
2465:
2461:
2456:
2452:
2448:
2445:
2391:
2390:
2379:
2372:
2368:
2362:
2358:
2354:
2349:
2345:
2339:
2335:
2331:
2326:
2322:
2316:
2312:
2308:
2303:
2299:
2295:
2290:
2286:
2282:
2277:
2273:
2267:
2264:
2250:
2249:
2238:
2233:
2230:
2227:
2222:
2218:
2214:
2209:
2205:
2201:
2196:
2192:
2184:
2181:
2160:
2157:
2137:
2136:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2063:
2059:
2053:
2049:
2045:
2040:
2036:
2030:
2026:
2022:
2017:
2013:
2007:
2003:
1979:
1976:
1975:
1974:
1973:
1972:
1961:
1950:
1946:
1942:
1939:
1934:
1930:
1923:
1920:
1917:
1910:
1906:
1902:
1897:
1893:
1887:
1884:
1871:
1870:
1859:
1858:
1857:
1856:
1841:
1834:
1828:
1825:
1822:
1819:
1814:
1809:
1806:
1803:
1800:
1794:
1788:
1785:
1780:
1777:
1775:
1773:
1770:
1769:
1765:
1759:
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1753:
1750:
1745:
1740:
1737:
1734:
1731:
1725:
1719:
1716:
1711:
1708:
1706:
1704:
1701:
1700:
1670:
1669:
1668:
1652:
1647:
1640:
1636:
1633:
1626:
1620:
1616:
1613:
1605:
1601:
1597:
1588:
1581:
1575:
1570:
1567:
1565:
1563:
1560:
1559:
1555:
1549:
1542:
1538:
1535:
1525:
1522:
1517:
1512:
1505:
1501:
1498:
1489:
1481:
1478:
1471:
1468:
1466:
1464:
1461:
1460:
1447:
1446:
1437:
1436:
1435:
1434:
1423:
1414:
1410:
1407:
1395:
1390:
1386:
1381:
1376:
1369:
1366:
1358:
1352:
1348:
1345:
1336:
1333:
1328:
1323:
1316:
1313:
1281:
1274:
1271:
1264:
1259:
1252:
1249:
1242:
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1229:
1222:
1219:
1212:
1207:
1201:
1194:
1191:
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1177:
1172:
1168:
1164:
1161:
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1155:
1140:
1133:
1132:
1131:
1120:
1117:
1114:
1111:
1106:
1102:
1098:
1092:
1089:
1082:
1079:
1074:
1071:
1066:
1061:
1057:
1053:
1047:
1044:
1037:
1034:
1029:
1025:
1021:
1015:
1012:
996:
995:
987:
986:
985:
984:
973:
967:
964:
961:
958:
955:
952:
948:
944:
941:
938:
932:
928:
925:
914:
907:
901:
895:
891:
888:
885:
879:
875:
872:
862:
859:
851:
845:
841:
838:
832:
829:
826:
823:
817:
814:
808:
802:
799:
783:
782:
761:
758:
733:
732:
704:
699:
695:
653:
648:
644:
638:
634:
628:
624:
620:
587:
583:
577:
573:
567:
563:
559:
509:
505:
501:
496:
492:
488:
483:
479:
449:
443:
428:
415:
409:
403:
391:
373:is called the
360:
356:
352:
347:
343:
339:
334:
330:
311:
310:
305:
302:
257:is the unique
219:
216:
213:
210:
207:
204:
201:
198:
194:
190:
187:
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178:
175:
169:
163:
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100:
97:
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62:
58:
54:
52:
15:
9:
6:
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3:
2:
3191:
3180:
3177:
3175:
3172:
3170:
3167:
3166:
3164:
3152:
3150:
3143:
3136:
3135:
3128:
3122:
3120:
3113:
3111:
3109:
3100:
3096:
3092:
3088:
3084:
3080:
3079:
3071:
3067:
3061:
3059:
3049:
3042:
3033:
3032:
3027:
3024:
3017:
3009:on 2012-08-26
3005:
3001:
2997:
2993:
2989:
2985:
2981:
2977:
2973:
2972:
2964:
2957:
2955:
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2934:
2928:
2926:
2924:
2919:
2911:
2877:
2874:
2862:
2859:
2853:
2844:
2841:
2827:
2824:
2818:
2809:
2806:
2797:
2785:
2782:
2776:
2767:
2764:
2750:
2747:
2741:
2732:
2729:
2720:
2708:
2705:
2699:
2690:
2687:
2673:
2670:
2664:
2655:
2652:
2639:
2638:
2637:
2626:
2619:
2616:
2614:
2610:
2605:
2591:
2580:
2576:
2572:
2564:
2554:
2550:
2546:
2519:
2514:
2510:
2506:
2501:
2497:
2467:
2463:
2459:
2454:
2450:
2446:
2443:
2433:
2432:Fermat points
2428:
2426:
2421:
2419:
2415:
2411:
2407:
2404:
2400:
2396:
2393:According to
2377:
2370:
2366:
2360:
2356:
2352:
2347:
2343:
2337:
2333:
2329:
2324:
2320:
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2301:
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2284:
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2275:
2271:
2265:
2262:
2255:
2254:
2253:
2236:
2231:
2228:
2225:
2220:
2216:
2212:
2207:
2203:
2199:
2194:
2190:
2182:
2179:
2170:
2169:
2168:
2156:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2061:
2057:
2051:
2047:
2043:
2038:
2034:
2028:
2024:
2020:
2015:
2011:
2005:
2001:
1993:
1992:
1991:
1985:
1959:
1948:
1944:
1940:
1937:
1932:
1928:
1921:
1918:
1915:
1908:
1904:
1900:
1895:
1891:
1885:
1882:
1875:
1874:
1873:
1872:
1865:
1861:
1860:
1839:
1832:
1826:
1823:
1820:
1817:
1812:
1807:
1804:
1801:
1798:
1792:
1786:
1783:
1778:
1776:
1771:
1763:
1757:
1754:
1751:
1748:
1743:
1738:
1735:
1732:
1729:
1723:
1717:
1714:
1709:
1707:
1702:
1691:
1690:
1687:
1683:
1675:
1671:
1650:
1645:
1634:
1631:
1624:
1614:
1611:
1603:
1595:
1586:
1579:
1573:
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1561:
1553:
1547:
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1533:
1523:
1520:
1515:
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1496:
1487:
1479:
1476:
1469:
1467:
1462:
1451:
1450:
1449:
1448:
1444:
1443:
1442:
1441:
1421:
1408:
1405:
1393:
1388:
1384:
1379:
1374:
1364:
1356:
1346:
1343:
1334:
1331:
1326:
1321:
1311:
1279:
1269:
1262:
1257:
1247:
1240:
1233:
1227:
1217:
1210:
1205:
1199:
1189:
1178:
1170:
1166:
1162:
1156:
1153:
1146:
1145:
1139:
1134:
1118:
1112:
1109:
1104:
1100:
1087:
1080:
1072:
1069:
1064:
1059:
1055:
1042:
1035:
1027:
1023:
1010:
1000:
999:
998:
997:
993:
989:
988:
971:
965:
962:
959:
956:
953:
950:
946:
942:
939:
936:
930:
926:
923:
912:
905:
899:
893:
889:
886:
883:
877:
873:
870:
860:
857:
849:
843:
839:
836:
830:
824:
812:
806:
797:
787:
786:
785:
784:
780:
775:
769:
768:
767:
766:
757:
755:
743:
736:of its image.
730:
729:
728:
726:
725:greatest area
702:
697:
693:
682:
676:
672:
668:
651:
646:
642:
636:
632:
626:
622:
609:
585:
581:
575:
571:
565:
561:
546:
537:
533:
527:
507:
503:
499:
494:
490:
486:
481:
477:
467:
460:
440:
435:
426:
400:
389:
385:
382:
376:
358:
354:
350:
345:
341:
337:
332:
328:
318:
308:
307:
301:
299:
295:
290:
288:
287:Jakob Steiner
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
244:
240:
214:
211:
208:
205:
202:
199:
192:
188:
185:
182:
176:
173:
167:
161:
158:
151:
147:
144:
141:
139:
128:
125:
122:
119:
116:
107:
104:
101:
98:
95:
92:
83:
80:
77:
74:
71:
68:
60:
56:
34:
26:
21:
3148:
3142:
3132:
3127:
3118:
3082:
3076:
3047:
3041:
3029:
3016:
3004:the original
2975:
2969:
2945:
2940:
2902:-gons: some
2897:
2624:
2620:
2617:
2606:
2429:
2422:
2392:
2251:
2162:
2138:
1981:
1863:
1685:
1681:
1439:
1438:
1137:
991:
773:
764:
763:
734:
724:
680:
674:
671:eccentricity
666:
607:
544:
531:
525:
465:
458:
457:
438:
398:
380:
374:
316:
312:
291:
250:
246:
242:
236:
41:(13/3, 11/3)
2944:H. Dörrie,
2412:, then the
2408:of a cubic
459:Properties:
283:equilateral
3163:Categories
2914:References
2609:Euler line
2418:derivative
2410:polynomial
1440:Semi-axes:
992:4 vertices
309:Definition
3066:Minda, D.
3031:MathWorld
2867:¯
2854:⋅
2849:¯
2832:¯
2819:⋅
2814:¯
2790:¯
2777:⋅
2772:¯
2755:¯
2742:⋅
2737:¯
2713:¯
2700:⋅
2695:¯
2678:¯
2665:⋅
2660:¯
2613:directrix
2565:±
2555:−
2515:−
2494:∠
2468:−
2353:−
2330:−
2307:−
2226:±
2103:−
2085:−
2067:−
1938:−
1919:⋯
1901:−
1821:−
1813:−
1752:−
1674:semi-axes
1639:→
1619:→
1541:→
1504:→
1413:→
1368:→
1351:→
1315:→
1273:→
1263:⋅
1251:→
1221:→
1211:−
1193:→
1157:
1113:π
1091:→
1070:π
1065:±
1046:→
1014:→
966:π
954:≤
940:
931:→
887:
878:→
844:→
816:→
801:→
754:Inellipse
694:π
619:△
558:△
524:a) There
271:inellipse
267:midpoints
212:−
186:−
145:−
126:−
105:−
81:−
2992:27642475
2636:we have
2399:vertices
742:incircle
536:centroid
298:centroid
255:triangle
239:geometry
43:, since
3099:2456092
3000:2398412
2611:as its
2532:
2486:
2482:
2436:
2403:complex
2165:a, b, c
1988:a, b, c
1679:(where
781: :
752:e) See
679:d) The
675:similar
530:b) The
263:tangent
259:ellipse
3097:
2998:
2990:
2910:-gon.
2252:where
2139:where
1957:
1837:
1419:
1135:where
969:
673:, are
667:scaled
532:center
526:exists
451:
445:
432:
430:
417:
411:
405:
393:
241:, the
37:(3, 5)
31:, the
3073:(PDF)
3007:(PDF)
2988:JSTOR
2966:(PDF)
2406:zeros
1684:>
1299:with
731:Proof
253:of a
249:, or
2632:and
2414:foci
2167:are
2151:and
1862:The
1677:a, b
990:The
960:<
681:area
277:and
39:and
33:foci
3087:doi
3083:115
2980:doi
2976:115
2625:ABC
1689:):
1600:det
1154:cot
937:sin
884:cos
774:ABC
608:ABC
545:ABC
541:of
466:ABC
439:ABC
399:ABC
381:ABC
377:of
317:ABC
237:In
3165::
3107:^
3095:MR
3093:,
3081:,
3075:,
3057:^
3028:.
2996:MR
2994:,
2986:,
2974:,
2968:,
2953:^
2922:^
2878:1.
2615:.
1569::=
1470::=
756:.
384:.
300:.
245:,
174:11
159:13
3102:.
3089::
3052:.
3034:.
3011:.
2982::
2935:.
2908:n
2904:n
2900:n
2875:=
2863:A
2860:C
2845:C
2842:B
2828:C
2825:Q
2810:C
2807:P
2798:+
2786:C
2783:B
2768:B
2765:A
2751:B
2748:Q
2733:B
2730:P
2721:+
2709:B
2706:A
2691:A
2688:C
2674:A
2671:Q
2656:A
2653:P
2634:Q
2630:P
2623:△
2592:;
2587:|
2581:+
2577:F
2573:G
2569:|
2561:|
2551:F
2547:G
2543:|
2520:.
2511:F
2507:G
2502:+
2498:F
2464:F
2460:,
2455:+
2451:F
2447:,
2444:G
2378:.
2371:2
2367:a
2361:2
2357:c
2348:2
2344:c
2338:2
2334:b
2325:2
2321:b
2315:2
2311:a
2302:4
2298:c
2294:+
2289:4
2285:b
2281:+
2276:4
2272:a
2266:=
2263:Z
2237:,
2232:Z
2229:2
2221:2
2217:c
2213:+
2208:2
2204:b
2200:+
2195:2
2191:a
2183:6
2180:1
2153:c
2149:b
2145:a
2141:x
2124:0
2121:=
2118:x
2115:z
2112:a
2109:c
2106:2
2100:z
2097:y
2094:c
2091:b
2088:2
2082:y
2079:x
2076:b
2073:a
2070:2
2062:2
2058:z
2052:2
2048:c
2044:+
2039:2
2035:y
2029:2
2025:b
2021:+
2016:2
2012:x
2006:2
2002:a
1960:.
1949:2
1945:N
1941:4
1933:2
1929:M
1922:=
1916:=
1909:2
1905:b
1896:2
1892:a
1886:=
1883:c
1867:c
1840:.
1833:)
1827:N
1824:2
1818:M
1808:N
1805:2
1802:+
1799:M
1793:(
1787:2
1784:1
1779:=
1772:b
1764:)
1758:N
1755:2
1749:M
1744:+
1739:N
1736:2
1733:+
1730:M
1724:(
1718:2
1715:1
1710:=
1703:a
1686:b
1682:a
1651:|
1646:)
1635:B
1632:A
1625:,
1615:C
1612:S
1604:(
1596:|
1587:3
1580:4
1574:1
1562:N
1554:)
1548:2
1537:B
1534:A
1524:3
1521:1
1516:+
1511:2
1500:C
1497:S
1488:(
1480:4
1477:1
1463:M
1422:.
1409:B
1406:A
1394:3
1389:2
1385:1
1380:=
1375:2
1365:f
1357:,
1347:C
1344:S
1335:2
1332:1
1327:=
1322:1
1312:f
1280:2
1270:f
1258:1
1248:f
1241:2
1234:2
1228:2
1218:f
1206:2
1200:1
1190:f
1179:=
1176:)
1171:0
1167:t
1163:2
1160:(
1141:0
1138:t
1119:,
1116:)
1110:+
1105:0
1101:t
1097:(
1088:p
1081:,
1078:)
1073:2
1060:0
1056:t
1052:(
1043:p
1036:,
1033:)
1028:0
1024:t
1020:(
1011:p
972:.
963:2
957:t
951:0
947:,
943:t
927:B
924:A
913:3
906:2
900:1
894:+
890:t
874:C
871:S
861:2
858:1
850:+
840:S
837:O
831:=
828:)
825:t
822:(
813:p
807:=
798:x
772:△
703:3
698:3
677:.
652:.
647:3
643:M
637:2
633:M
627:1
623:M
606:△
602:S
586:3
582:M
576:2
572:M
566:1
562:M
548:.
543:△
539:S
508:3
504:M
500:,
495:2
491:M
487:,
482:1
478:M
464:△
437:△
397:△
379:△
359:3
355:M
351:,
346:2
342:M
338:,
333:1
329:M
315:△
218:)
215:x
209:i
206:5
203:+
200:3
197:(
193:)
189:x
183:i
177:3
168:+
162:3
152:(
148:3
142:=
132:)
129:x
123:i
120:+
117:3
114:(
111:)
108:x
102:i
99:5
96:+
93:7
90:(
87:)
84:x
78:i
75:7
72:+
69:1
66:(
61:x
57:D
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