22:
169:, each at some fixed distance from the polygon; the medial axis is traced out by the vertices of these curves. In the case of a triangle, the medial axis consists of three segments of the angle bisectors, connecting the vertices of the triangle to the incenter, which is the unique point on the innermost offset curve. The
1916:
2836:
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter; every line through the incenter that splits the area in half also splits the perimeter in half. There are either one, two, or three of these lines for any given triangle.
1085:
A line that is an angle bisector is equidistant from both of its lines when measuring by the perpendicular. At the point where two bisectors intersect, this point is perpendicularly equidistant from the final angle's forming lines (because they are the same distance from this angles opposite edge),
144:
The incenter lies at equal distances from the three line segments forming the sides of the triangle, and also from the three lines containing those segments. It is the only point equally distant from the line segments, but there are three more points equally distant from the lines, the excenters,
1415:
of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter lies inside the
141:, Proposition 4 of Book IV proves that this point is also the center of the inscribed circle of the triangle. The incircle itself may be constructed by dropping a perpendicular from the incenter to one of the sides of the triangle and drawing a circle with that segment as its radius.
2078:
1620:
2742:
1010:
924:
835:
711:
2570:
160:
of a polygon is the set of points whose nearest neighbor on the polygon is not unique: these points are equidistant from two or more sides of the polygon. One method for computing medial axes is using the
2364:
2469:
1940:
2147:
1338:
2826:
2784:
25:
The point of intersection of angle bisectors of the 3 angles of triangle ABC is the incenter (denoted by I). The incircle (whose center is I) touches each side of the triangle.
2898:
1045:
560:
503:
471:
414:
382:
330:
247:
41:, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal
2262:
749:
625:
1911:{\displaystyle {\bigg (}{\frac {ax_{A}+bx_{B}+cx_{C}}{a+b+c}},{\frac {ay_{A}+by_{B}+cy_{C}}{a+b+c}}{\bigg )}={\frac {a(x_{A},y_{A})+b(x_{B},y_{B})+c(x_{C},y_{C})}{a+b+c}}.}
1552:
1506:
1460:
1164:
for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions. Barycentric coordinates for the incenter are given by
1076:
591:
298:
215:
2583:(the triangle whose vertices are the midpoints of the sides) and therefore lies inside this triangle. Conversely the Nagel point of any triangle is the incenter of its
3343:
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
1139:
1200:
173:, defined in a similar way from a different type of offset curve, coincides with the medial axis for convex polygons and so also has its junction at the incenter.
525:
436:
1612:
1592:
1572:
1401:
1381:
1361:
1263:
1243:
1223:
350:
267:
2678:
929:
843:
754:
630:
2480:
1161:
146:
58:
3238:
3215:
76:, it is one of the four triangle centers known to the ancient Greeks, and the only one of the four that does not in general lie on the
3292:
2966:
2275:
are the circumradius and the inradius respectively; thus the circumradius is at least twice the inradius, with equality only in the
1103:
for a point in the triangle give the ratio of distances to the triangle sides. Trilinear coordinates for the incenter are given by
3134:
2296:
85:
2382:
2193:
111:
to each side of the polygon. In this case the incenter is the center of this circle and is equally distant from all sides.
2073:{\displaystyle {\frac {IA\cdot IA}{CA\cdot AB}}+{\frac {IB\cdot IB}{AB\cdot BC}}+{\frac {IC\cdot IC}{BC\cdot CA}}=1.}
1934:, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation
2584:
3465:
2089:
3085:
Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity",
2645:, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
3018:
Blum, Harry (1967), "A transformation for extracting new descriptors of shape", in Wathen-Dunn, Weiant (ed.),
2641:, among other points. The incenter generally does not lie on the Euler line; it is on the Euler line only for
1275:
3114:
3404:
Arie
Bialostocki and Dora Bialostocki, "The incenter and an excenter as solutions to an extremal problem",
3029:
In the triangle three corners start propagating and disappear at the center of the largest inscribed circle
2792:
2750:
3309:
3230:
Marie-Nicole Gras, "Distances between the circumcenter of the extouch triangle and the classical centers"
714:
3278:
1412:
1100:
2864:
3307:
Edmonds, Allan L.; Hajja, Mowaffaq; Martini, Horst (2008), "Orthocentric simplices and biregularity",
1018:
533:
476:
444:
387:
355:
303:
220:
3370:
3087:
3019:
2989:
2982:
2606:
2276:
2210:
723:
599:
136:
93:
3207:
Dragutin Svrtan and Darko Veljan, "Non-Euclidean versions of some classical triangle inequalities",
1148:
under coordinatewise multiplication of trilinear coordinates; in this group, the incenter forms the
1511:
1465:
1419:
1145:
162:
150:
54:
1050:
565:
272:
189:
3387:
3334:
2933:
2642:
2591:
124:
104:
3438:
3288:
3043:
2610:
2287:
2186:
1109:
170:
3409:
3235:
3212:
1170:
3379:
3318:
3282:
3096:
3059:
2925:
1149:
89:
81:
3330:
3153:
3071:
2945:
3326:
3242:
3219:
3149:
3067:
2970:
2963:
2941:
2580:
38:
2916:
Kimberling, Clark (1994), "Central Points and
Central Lines in the Plane of a Triangle",
2617:). Any other point within the orthocentroidal disk is the incenter of a unique triangle.
508:
419:
3420:
Hajja, Mowaffaq, Extremal properties of the incentre and the excenters of a triangle",
3157:
1597:
1577:
1557:
1386:
1366:
1346:
1248:
1228:
1208:
335:
252:
166:
128:
42:
3459:
3391:
3338:
2737:{\displaystyle {\frac {d}{s}}<{\frac {d}{u}}<{\frac {d}{v}}<{\frac {1}{3}};}
2929:
2664:, the length of the Euler line segment from the orthocenter to the circumcenter as
2630:
2161:
1266:
1005:{\displaystyle {\overline {AC}}:{\overline {BC}}={\overline {AF}}:{\overline {BF}}}
919:{\displaystyle {\overline {AC}}:{\overline {AF}}={\overline {BC}}:{\overline {BF}}}
830:{\displaystyle {\overline {BC}}:{\overline {BF}}={\overline {CI}}:{\overline {IF}}}
706:{\displaystyle {\overline {AC}}:{\overline {AF}}={\overline {CI}}:{\overline {IF}}}
69:
3063:
3047:
2638:
2599:
2576:
2370:
157:
73:
50:
46:
3441:
3322:
3100:
2626:
77:
3383:
3446:
2634:
2182:
2165:
1265:
are the lengths of the sides of the triangle, or equivalently (using the
65:
30:
3284:
Geometry Turned On: Dynamic
Software in Learning, Teaching, and Research
2937:
2613:, whose position is fixed 1/4 of the way along the diameter (closer to
2565:{\displaystyle IG<HG,\quad IH<HG,\quad IG<IO,\quad 2IN<IO.}
120:
108:
100:
132:
21:
1554:, and the sides opposite these vertices have corresponding lengths
1144:
The collection of triangle centers may be given the structure of a
149:
of the given triangle. The incenter and excenters together form an
1416:
triangle as stated above.) If the three vertices are located at
2990:
Book IV, Proposition 4: To inscribe a circle in a given triangle
2648:
Denoting the distance from the incenter to the Euler line as
3368:
Kodokostas, Dimitrios (April 2010), "Triangle equalizers",
2359:{\displaystyle IN={\frac {1}{2}}(R-2r)<{\frac {1}{2}}R.}
3410:
http://forumgeom.fau.edu/FG2011volume11/FG201102index.html
3236:
http://forumgeom.fau.edu/FG2014volume14/FG201405index.html
3213:
http://forumgeom.fau.edu/FG2012volume12/FG201217index.html
3287:, The Mathematical Association of America, pp. 3–4,
103:
with more than three sides, the incenter only exists for
49:
from the triangle's sides, as the junction point of the
2464:{\displaystyle IH^{2}=2r^{2}-4R^{2}\cos A\cos B\cos C.}
2992:. David Joyce, Clark University, retrieved 2014-10-28.
2869:
2849:
be a variable point on the internal angle bisector of
2867:
2795:
2753:
2681:
2483:
2385:
2299:
2213:
2092:
1943:
1623:
1600:
1580:
1560:
1514:
1468:
1422:
1389:
1369:
1349:
1278:
1251:
1231:
1211:
1173:
1112:
1053:
1021:
932:
846:
757:
726:
633:
602:
568:
536:
511:
479:
447:
422:
390:
358:
338:
306:
275:
255:
223:
192:
3021:
Models for the
Perception of Speech and Visual Form
3135:"The distance from the incenter to the Euler line"
2892:
2820:
2778:
2736:
2564:
2463:
2358:
2256:
2141:
2072:
1910:
1606:
1586:
1566:
1546:
1500:
1454:
1395:
1375:
1355:
1332:
1257:
1237:
1217:
1194:
1133:
1070:
1039:
1004:
918:
829:
743:
705:
619:
585:
554:
519:
497:
465:
430:
408:
376:
344:
324:
292:
261:
241:
209:
2185:is less than one third the length of the longest
1772:
1626:
2861:(the incenter) maximizes or minimizes the ratio
57:of the triangle, and as the center point of the
1086:and therefore lies on its angle bisector line.
2369:The squared distance from the incenter to the
165:, in which one forms a continuous sequence of
2282:The distance from the incenter to the center
8:
2629:of a triangle is a line passing through its
16:Center of the inscribed circle of a triangle
2590:The incenter must lie in the interior of a
2142:{\displaystyle IA\cdot IB\cdot IC=4Rr^{2},}
3027:, Cambridge: MIT Press, pp. 362–380,
80:. It is the first listed center, X(1), in
3046:; Alberts, David; Gärtner, Bernd (1995),
2868:
2866:
2841:Relative distances from an angle bisector
2802:
2794:
2760:
2752:
2721:
2708:
2695:
2682:
2680:
2482:
2425:
2409:
2393:
2384:
2340:
2309:
2298:
2221:
2212:
2196:, the squared distance from the incenter
2130:
2091:
2026:
1985:
1944:
1942:
1876:
1863:
1841:
1828:
1806:
1793:
1780:
1771:
1770:
1744:
1728:
1712:
1702:
1673:
1657:
1641:
1631:
1625:
1624:
1622:
1599:
1579:
1559:
1535:
1522:
1513:
1489:
1476:
1467:
1443:
1430:
1421:
1388:
1368:
1348:
1277:
1250:
1230:
1210:
1172:
1111:
1057:
1052:
1022:
1020:
987:
969:
951:
933:
931:
901:
883:
865:
847:
845:
812:
794:
776:
758:
756:
730:
725:
688:
670:
652:
634:
632:
606:
601:
572:
567:
537:
535:
512:
510:
480:
478:
448:
446:
423:
421:
391:
389:
359:
357:
337:
307:
305:
279:
274:
254:
224:
222:
196:
191:
131:of a triangle meet in a single point. In
3355:
3265:
3253:
3195:
20:
3191:
3189:
3180:
3048:"A novel type of skeleton for polygons"
3006:, Boston: Houghton Mifflin, p. 182
2908:
1333:{\displaystyle \sin(A):\sin(B):\sin(C)}
1090:Relation to triangle sides and vertices
2652:, the length of the longest median as
2181:The distance from the incenter to the
1403:are the angles at the three vertices.
107:- those that have an incircle that is
3052:Journal of Universal Computer Science
2959:
2957:
2955:
2821:{\displaystyle d<{\frac {1}{2}}R.}
2779:{\displaystyle d<{\frac {1}{3}}e;}
2594:whose diameter connects the centroid
7:
2656:, the length of the longest side as
45:of the triangle cross, as the point
2672:, the following inequalities hold:
2609:), but it cannot coincide with the
1926:Denoting the incenter of triangle
1054:
727:
603:
569:
276:
193:
14:
2893:{\displaystyle {\tfrac {BX}{CX}}}
2964:Encyclopedia of Triangle Centers
1040:{\displaystyle {\overline {CF}}}
555:{\displaystyle {\overline {CI}}}
498:{\displaystyle {\overline {AB}}}
466:{\displaystyle {\overline {CI}}}
409:{\displaystyle {\overline {BE}}}
377:{\displaystyle {\overline {AD}}}
325:{\displaystyle {\overline {AC}}}
242:{\displaystyle {\overline {BC}}}
86:Encyclopedia of Triangle Centers
2540:
2521:
2502:
2257:{\displaystyle OI^{2}=R(R-2r),}
744:{\displaystyle \triangle {BCF}}
620:{\displaystyle \triangle {ACF}}
2930:10.1080/0025570X.1994.11996210
2334:
2319:
2248:
2233:
1882:
1856:
1847:
1821:
1812:
1786:
1541:
1515:
1495:
1469:
1449:
1423:
1327:
1321:
1309:
1303:
1291:
1285:
145:which form the centers of the
1:
3256:, Lemma 1, p. 233.
3133:Franzsen, William N. (2011),
1547:{\displaystyle (x_{C},y_{C})}
1501:{\displaystyle (x_{B},y_{B})}
1455:{\displaystyle (x_{A},y_{A})}
3064:10.1007/978-3-642-80350-5_65
2832:Area and perimeter splitters
1071:{\displaystyle \angle {ACB}}
1032:
997:
979:
961:
943:
911:
893:
875:
857:
822:
804:
786:
768:
698:
680:
662:
644:
586:{\displaystyle \angle {ACB}}
547:
490:
458:
401:
369:
317:
293:{\displaystyle \angle {ABC}}
234:
210:{\displaystyle \angle {BAC}}
2900:along that angle bisector.
2668:, and the semiperimeter as
2194:Euler's theorem in geometry
530:Then we have to prove that
115:Definition and construction
53:and innermost point of the
3482:
2585:anticomplementary triangle
1614:, then the incenter is at
3323:10.1007/s00025-008-0294-4
3101:10.1017/S0025557200004277
3384:10.4169/002557010X482916
3115:Altshiller-Court, Nathan
1134:{\displaystyle \ 1:1:1.}
127:that the three interior
3424:96, July 2012, 315-317.
3171:. Lemma 3, p. 233.
3002:Johnson, R. A. (1929),
1195:{\displaystyle \ a:b:c}
1162:barycentric coordinates
1156:Barycentric coordinates
269:, and the bisection of
3310:Results in Mathematics
3281:; King, James (1997),
3279:Schattschneider, Doris
3211:12 (2012), 197–209.
2973:, accessed 2014-10-28.
2894:
2822:
2780:
2738:
2660:, the circumradius as
2566:
2474:Inequalities include:
2465:
2360:
2258:
2143:
2074:
1912:
1608:
1588:
1568:
1548:
1502:
1456:
1397:
1377:
1357:
1334:
1259:
1239:
1219:
1196:
1135:
1072:
1041:
1006:
920:
831:
745:
715:Angle bisector theorem
707:
621:
587:
556:
521:
499:
467:
432:
410:
378:
346:
326:
294:
263:
243:
211:
26:
2895:
2823:
2781:
2739:
2567:
2466:
2361:
2259:
2172:Related constructions
2144:
2075:
1922:Distances to vertices
1913:
1609:
1589:
1569:
1549:
1503:
1457:
1413:Cartesian coordinates
1407:Cartesian coordinates
1398:
1378:
1358:
1335:
1260:
1240:
1220:
1197:
1136:
1101:trilinear coordinates
1095:Trilinear coordinates
1073:
1042:
1007:
921:
832:
746:
708:
622:
588:
557:
522:
500:
468:
433:
411:
379:
347:
327:
295:
264:
244:
212:
186:Let the bisection of
96:of triangle centers.
24:
3422:Mathematical Gazette
3371:Mathematics Magazine
3121:, Dover Publications
3088:Mathematical Gazette
2918:Mathematics Magazine
2865:
2793:
2751:
2679:
2607:orthocentroidal disk
2575:The incenter is the
2481:
2383:
2297:
2211:
2200:to the circumcenter
2090:
1941:
1621:
1598:
1578:
1558:
1512:
1466:
1420:
1387:
1367:
1347:
1276:
1249:
1229:
1209:
1171:
1110:
1051:
1047:is the bisection of
1019:
930:
844:
755:
724:
631:
600:
566:
562:is the bisection of
534:
509:
477:
445:
420:
388:
356:
336:
304:
273:
253:
221:
190:
94:multiplicative group
3406:Forum Geometricorum
3358:, pp. 232–234.
3234:14 (2014), 51-61.
3232:Forum Geometricorum
3209:Forum Geometricorum
3142:Forum Geometricorum
3123:. #84, p. 121.
3042:Aichholzer, Oswin;
2643:isosceles triangles
2160:are the triangle's
1081:Perpendicular proof
520:{\displaystyle {F}}
431:{\displaystyle {I}}
163:grassfire transform
151:orthocentric system
105:tangential polygons
55:grassfire transform
37:of a triangle is a
3439:Weisstein, Eric W.
3408:11 (2011), 9-12.
3241:2021-04-28 at the
3218:2019-10-28 at the
3044:Aurenhammer, Franz
2969:2012-04-19 at the
2890:
2888:
2818:
2776:
2734:
2562:
2461:
2356:
2254:
2139:
2070:
1908:
1604:
1584:
1564:
1544:
1498:
1452:
1393:
1373:
1353:
1330:
1255:
1235:
1215:
1192:
1131:
1068:
1037:
1002:
916:
827:
741:
703:
617:
583:
552:
517:
495:
463:
428:
406:
374:
342:
322:
290:
259:
239:
207:
125:Euclidean geometry
64:Together with the
27:
2887:
2810:
2768:
2729:
2716:
2703:
2690:
2611:nine-point center
2348:
2317:
2288:nine point circle
2189:of the triangle.
2062:
2021:
1980:
1903:
1768:
1697:
1607:{\displaystyle c}
1587:{\displaystyle b}
1567:{\displaystyle a}
1396:{\displaystyle C}
1376:{\displaystyle B}
1356:{\displaystyle A}
1258:{\displaystyle c}
1238:{\displaystyle b}
1218:{\displaystyle a}
1176:
1115:
1035:
1000:
982:
964:
946:
914:
896:
878:
860:
825:
807:
789:
771:
701:
683:
665:
647:
550:
493:
461:
404:
372:
345:{\displaystyle E}
320:
262:{\displaystyle D}
237:
171:straight skeleton
61:of the triangle.
3473:
3466:Triangle centers
3452:
3451:
3425:
3418:
3412:
3402:
3396:
3394:
3365:
3359:
3353:
3347:
3345:
3304:
3298:
3297:
3275:
3269:
3263:
3257:
3251:
3245:
3228:
3222:
3205:
3199:
3193:
3184:
3178:
3172:
3170:
3169:
3168:
3162:
3156:, archived from
3139:
3130:
3124:
3122:
3119:College Geometry
3111:
3105:
3103:
3095:(535): 161–165,
3082:
3076:
3074:
3039:
3033:
3031:
3026:
3015:
3009:
3007:
2999:
2993:
2980:
2974:
2961:
2950:
2948:
2913:
2899:
2897:
2896:
2891:
2889:
2886:
2878:
2870:
2827:
2825:
2824:
2819:
2811:
2803:
2785:
2783:
2782:
2777:
2769:
2761:
2743:
2741:
2740:
2735:
2730:
2722:
2717:
2709:
2704:
2696:
2691:
2683:
2571:
2569:
2568:
2563:
2470:
2468:
2467:
2462:
2430:
2429:
2414:
2413:
2398:
2397:
2365:
2363:
2362:
2357:
2349:
2341:
2318:
2310:
2263:
2261:
2260:
2255:
2226:
2225:
2148:
2146:
2145:
2140:
2135:
2134:
2079:
2077:
2076:
2071:
2063:
2061:
2044:
2027:
2022:
2020:
2003:
1986:
1981:
1979:
1962:
1945:
1917:
1915:
1914:
1909:
1904:
1902:
1885:
1881:
1880:
1868:
1867:
1846:
1845:
1833:
1832:
1811:
1810:
1798:
1797:
1781:
1776:
1775:
1769:
1767:
1750:
1749:
1748:
1733:
1732:
1717:
1716:
1703:
1698:
1696:
1679:
1678:
1677:
1662:
1661:
1646:
1645:
1632:
1630:
1629:
1613:
1611:
1610:
1605:
1593:
1591:
1590:
1585:
1573:
1571:
1570:
1565:
1553:
1551:
1550:
1545:
1540:
1539:
1527:
1526:
1507:
1505:
1504:
1499:
1494:
1493:
1481:
1480:
1461:
1459:
1458:
1453:
1448:
1447:
1435:
1434:
1402:
1400:
1399:
1394:
1382:
1380:
1379:
1374:
1362:
1360:
1359:
1354:
1339:
1337:
1336:
1331:
1264:
1262:
1261:
1256:
1244:
1242:
1241:
1236:
1224:
1222:
1221:
1216:
1201:
1199:
1198:
1193:
1174:
1150:identity element
1140:
1138:
1137:
1132:
1113:
1077:
1075:
1074:
1069:
1067:
1046:
1044:
1043:
1038:
1036:
1031:
1023:
1011:
1009:
1008:
1003:
1001:
996:
988:
983:
978:
970:
965:
960:
952:
947:
942:
934:
925:
923:
922:
917:
915:
910:
902:
897:
892:
884:
879:
874:
866:
861:
856:
848:
836:
834:
833:
828:
826:
821:
813:
808:
803:
795:
790:
785:
777:
772:
767:
759:
750:
748:
747:
742:
740:
712:
710:
709:
704:
702:
697:
689:
684:
679:
671:
666:
661:
653:
648:
643:
635:
626:
624:
623:
618:
616:
592:
590:
589:
584:
582:
561:
559:
558:
553:
551:
546:
538:
526:
524:
523:
518:
516:
504:
502:
501:
496:
494:
489:
481:
472:
470:
469:
464:
462:
457:
449:
437:
435:
434:
429:
427:
415:
413:
412:
407:
405:
400:
392:
383:
381:
380:
375:
373:
368:
360:
351:
349:
348:
343:
331:
329:
328:
323:
321:
316:
308:
299:
297:
296:
291:
289:
268:
266:
265:
260:
248:
246:
245:
240:
238:
233:
225:
216:
214:
213:
208:
206:
90:identity element
82:Clark Kimberling
59:inscribed circle
3481:
3480:
3476:
3475:
3474:
3472:
3471:
3470:
3456:
3455:
3437:
3436:
3433:
3428:
3419:
3415:
3403:
3399:
3367:
3366:
3362:
3356:Franzsen (2011)
3354:
3350:
3306:
3305:
3301:
3295:
3277:
3276:
3272:
3266:Franzsen (2011)
3264:
3260:
3254:Franzsen (2011)
3252:
3248:
3243:Wayback Machine
3229:
3225:
3220:Wayback Machine
3206:
3202:
3198:, p. 232.
3196:Franzsen (2011)
3194:
3187:
3179:
3175:
3166:
3164:
3160:
3137:
3132:
3131:
3127:
3113:
3112:
3108:
3084:
3083:
3079:
3058:(12): 752–761,
3041:
3040:
3036:
3024:
3017:
3016:
3012:
3004:Modern Geometry
3001:
3000:
2996:
2981:
2977:
2971:Wayback Machine
2962:
2953:
2915:
2914:
2910:
2906:
2879:
2871:
2863:
2862:
2843:
2834:
2791:
2790:
2749:
2748:
2677:
2676:
2623:
2581:medial triangle
2479:
2478:
2421:
2405:
2389:
2381:
2380:
2295:
2294:
2217:
2209:
2208:
2179:
2174:
2126:
2088:
2087:
2045:
2028:
2004:
1987:
1963:
1946:
1939:
1938:
1924:
1886:
1872:
1859:
1837:
1824:
1802:
1789:
1782:
1751:
1740:
1724:
1708:
1704:
1680:
1669:
1653:
1637:
1633:
1619:
1618:
1596:
1595:
1576:
1575:
1556:
1555:
1531:
1518:
1510:
1509:
1485:
1472:
1464:
1463:
1439:
1426:
1418:
1417:
1409:
1385:
1384:
1365:
1364:
1345:
1344:
1274:
1273:
1247:
1246:
1227:
1226:
1207:
1206:
1169:
1168:
1158:
1108:
1107:
1097:
1092:
1083:
1049:
1048:
1024:
1017:
1016:
989:
971:
953:
935:
928:
927:
903:
885:
867:
849:
842:
841:
814:
796:
778:
760:
753:
752:
722:
721:
690:
672:
654:
636:
629:
628:
598:
597:
564:
563:
539:
532:
531:
507:
506:
482:
475:
474:
450:
443:
442:
418:
417:
393:
386:
385:
361:
354:
353:
334:
333:
309:
302:
301:
271:
270:
251:
250:
226:
219:
218:
188:
187:
184:
179:
129:angle bisectors
117:
43:angle bisectors
39:triangle center
17:
12:
11:
5:
3479:
3477:
3469:
3468:
3458:
3457:
3454:
3453:
3432:
3431:External links
3429:
3427:
3426:
3413:
3397:
3378:(2): 141–146,
3360:
3348:
3317:(1–2): 41–50,
3299:
3294:978-0883850992
3293:
3270:
3268:, p. 232.
3258:
3246:
3223:
3200:
3185:
3181:Johnson (1929)
3173:
3125:
3106:
3077:
3034:
3010:
2994:
2975:
2951:
2924:(3): 163–187,
2907:
2905:
2902:
2885:
2882:
2877:
2874:
2842:
2839:
2833:
2830:
2829:
2828:
2817:
2814:
2809:
2806:
2801:
2798:
2787:
2786:
2775:
2772:
2767:
2764:
2759:
2756:
2745:
2744:
2733:
2728:
2725:
2720:
2715:
2712:
2707:
2702:
2699:
2694:
2689:
2686:
2622:
2619:
2573:
2572:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2539:
2536:
2533:
2530:
2527:
2524:
2520:
2517:
2514:
2511:
2508:
2505:
2501:
2498:
2495:
2492:
2489:
2486:
2472:
2471:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2428:
2424:
2420:
2417:
2412:
2408:
2404:
2401:
2396:
2392:
2388:
2367:
2366:
2355:
2352:
2347:
2344:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2316:
2313:
2308:
2305:
2302:
2265:
2264:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2224:
2220:
2216:
2178:
2175:
2173:
2170:
2168:respectively.
2150:
2149:
2138:
2133:
2129:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2083:Additionally,
2081:
2080:
2069:
2066:
2060:
2057:
2054:
2051:
2048:
2043:
2040:
2037:
2034:
2031:
2025:
2019:
2016:
2013:
2010:
2007:
2002:
1999:
1996:
1993:
1990:
1984:
1978:
1975:
1972:
1969:
1966:
1961:
1958:
1955:
1952:
1949:
1923:
1920:
1919:
1918:
1907:
1901:
1898:
1895:
1892:
1889:
1884:
1879:
1875:
1871:
1866:
1862:
1858:
1855:
1852:
1849:
1844:
1840:
1836:
1831:
1827:
1823:
1820:
1817:
1814:
1809:
1805:
1801:
1796:
1792:
1788:
1785:
1779:
1774:
1766:
1763:
1760:
1757:
1754:
1747:
1743:
1739:
1736:
1731:
1727:
1723:
1720:
1715:
1711:
1707:
1701:
1695:
1692:
1689:
1686:
1683:
1676:
1672:
1668:
1665:
1660:
1656:
1652:
1649:
1644:
1640:
1636:
1628:
1603:
1583:
1563:
1543:
1538:
1534:
1530:
1525:
1521:
1517:
1497:
1492:
1488:
1484:
1479:
1475:
1471:
1451:
1446:
1442:
1438:
1433:
1429:
1425:
1408:
1405:
1392:
1372:
1352:
1341:
1340:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1254:
1234:
1214:
1203:
1202:
1191:
1188:
1185:
1182:
1179:
1157:
1154:
1142:
1141:
1130:
1127:
1124:
1121:
1118:
1096:
1093:
1091:
1088:
1082:
1079:
1066:
1063:
1060:
1056:
1034:
1030:
1027:
999:
995:
992:
986:
981:
977:
974:
968:
963:
959:
956:
950:
945:
941:
938:
913:
909:
906:
900:
895:
891:
888:
882:
877:
873:
870:
864:
859:
855:
852:
824:
820:
817:
811:
806:
802:
799:
793:
788:
784:
781:
775:
770:
766:
763:
739:
736:
733:
729:
700:
696:
693:
687:
682:
678:
675:
669:
664:
660:
657:
651:
646:
642:
639:
615:
612:
609:
605:
581:
578:
575:
571:
549:
545:
542:
515:
492:
488:
485:
460:
456:
453:
426:
403:
399:
396:
371:
367:
364:
341:
319:
315:
312:
288:
285:
282:
278:
258:
236:
232:
229:
205:
202:
199:
195:
183:
180:
178:
175:
116:
113:
15:
13:
10:
9:
6:
4:
3:
2:
3478:
3467:
3464:
3463:
3461:
3449:
3448:
3443:
3440:
3435:
3434:
3430:
3423:
3417:
3414:
3411:
3407:
3401:
3398:
3393:
3389:
3385:
3381:
3377:
3373:
3372:
3364:
3361:
3357:
3352:
3349:
3344:
3340:
3336:
3332:
3328:
3324:
3320:
3316:
3312:
3311:
3303:
3300:
3296:
3290:
3286:
3285:
3280:
3274:
3271:
3267:
3262:
3259:
3255:
3250:
3247:
3244:
3240:
3237:
3233:
3227:
3224:
3221:
3217:
3214:
3210:
3204:
3201:
3197:
3192:
3190:
3186:
3183:, p. 186
3182:
3177:
3174:
3163:on 2020-12-05
3159:
3155:
3151:
3147:
3143:
3136:
3129:
3126:
3120:
3116:
3110:
3107:
3102:
3098:
3094:
3090:
3089:
3081:
3078:
3073:
3069:
3065:
3061:
3057:
3053:
3049:
3045:
3038:
3035:
3030:
3023:
3022:
3014:
3011:
3005:
2998:
2995:
2991:
2987:
2986:
2979:
2976:
2972:
2968:
2965:
2960:
2958:
2956:
2952:
2947:
2943:
2939:
2935:
2931:
2927:
2923:
2919:
2912:
2909:
2903:
2901:
2883:
2880:
2875:
2872:
2860:
2856:
2852:
2848:
2840:
2838:
2831:
2815:
2812:
2807:
2804:
2799:
2796:
2789:
2788:
2773:
2770:
2765:
2762:
2757:
2754:
2747:
2746:
2731:
2726:
2723:
2718:
2713:
2710:
2705:
2700:
2697:
2692:
2687:
2684:
2675:
2674:
2673:
2671:
2667:
2663:
2659:
2655:
2651:
2646:
2644:
2640:
2636:
2632:
2628:
2620:
2618:
2616:
2612:
2608:
2604:
2601:
2597:
2593:
2588:
2586:
2582:
2578:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2537:
2534:
2531:
2528:
2525:
2522:
2518:
2515:
2512:
2509:
2506:
2503:
2499:
2496:
2493:
2490:
2487:
2484:
2477:
2476:
2475:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2426:
2422:
2418:
2415:
2410:
2406:
2402:
2399:
2394:
2390:
2386:
2379:
2378:
2377:
2375:
2372:
2353:
2350:
2345:
2342:
2337:
2331:
2328:
2325:
2322:
2314:
2311:
2306:
2303:
2300:
2293:
2292:
2291:
2289:
2285:
2280:
2278:
2274:
2270:
2251:
2245:
2242:
2239:
2236:
2230:
2227:
2222:
2218:
2214:
2207:
2206:
2205:
2203:
2199:
2195:
2190:
2188:
2184:
2177:Other centers
2176:
2171:
2169:
2167:
2163:
2159:
2155:
2136:
2131:
2127:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2086:
2085:
2084:
2067:
2064:
2058:
2055:
2052:
2049:
2046:
2041:
2038:
2035:
2032:
2029:
2023:
2017:
2014:
2011:
2008:
2005:
2000:
1997:
1994:
1991:
1988:
1982:
1976:
1973:
1970:
1967:
1964:
1959:
1956:
1953:
1950:
1947:
1937:
1936:
1935:
1933:
1929:
1921:
1905:
1899:
1896:
1893:
1890:
1887:
1877:
1873:
1869:
1864:
1860:
1853:
1850:
1842:
1838:
1834:
1829:
1825:
1818:
1815:
1807:
1803:
1799:
1794:
1790:
1783:
1777:
1764:
1761:
1758:
1755:
1752:
1745:
1741:
1737:
1734:
1729:
1725:
1721:
1718:
1713:
1709:
1705:
1699:
1693:
1690:
1687:
1684:
1681:
1674:
1670:
1666:
1663:
1658:
1654:
1650:
1647:
1642:
1638:
1634:
1617:
1616:
1615:
1601:
1581:
1561:
1536:
1532:
1528:
1523:
1519:
1490:
1486:
1482:
1477:
1473:
1444:
1440:
1436:
1431:
1427:
1414:
1406:
1404:
1390:
1370:
1350:
1324:
1318:
1315:
1312:
1306:
1300:
1297:
1294:
1288:
1282:
1279:
1272:
1271:
1270:
1268:
1252:
1232:
1212:
1189:
1186:
1183:
1180:
1177:
1167:
1166:
1165:
1163:
1155:
1153:
1151:
1147:
1128:
1125:
1122:
1119:
1116:
1106:
1105:
1104:
1102:
1094:
1089:
1087:
1080:
1078:
1064:
1061:
1058:
1028:
1025:
1013:
993:
990:
984:
975:
972:
966:
957:
954:
948:
939:
936:
907:
904:
898:
889:
886:
880:
871:
868:
862:
853:
850:
838:
818:
815:
809:
800:
797:
791:
782:
779:
773:
764:
761:
737:
734:
731:
718:
716:
694:
691:
685:
676:
673:
667:
658:
655:
649:
640:
637:
613:
610:
607:
594:
579:
576:
573:
543:
540:
528:
513:
486:
483:
454:
451:
439:
424:
397:
394:
365:
362:
339:
313:
310:
286:
283:
280:
256:
230:
227:
203:
200:
197:
181:
176:
174:
172:
168:
167:offset curves
164:
159:
154:
152:
148:
142:
140:
139:
134:
130:
126:
122:
114:
112:
110:
106:
102:
97:
95:
91:
87:
83:
79:
75:
71:
67:
62:
60:
56:
52:
48:
44:
40:
36:
32:
23:
19:
3445:
3421:
3416:
3405:
3400:
3375:
3369:
3363:
3351:
3342:
3314:
3308:
3302:
3283:
3273:
3261:
3249:
3231:
3226:
3208:
3203:
3176:
3165:, retrieved
3158:the original
3145:
3141:
3128:
3118:
3109:
3092:
3086:
3080:
3055:
3051:
3037:
3028:
3020:
3013:
3003:
2997:
2984:
2978:
2921:
2917:
2911:
2858:
2854:
2850:
2846:
2844:
2835:
2669:
2665:
2661:
2657:
2653:
2649:
2647:
2631:circumcenter
2624:
2614:
2602:
2595:
2589:
2574:
2473:
2373:
2368:
2283:
2281:
2272:
2268:
2266:
2204:is given by
2201:
2197:
2191:
2180:
2162:circumradius
2157:
2153:
2151:
2082:
1931:
1927:
1925:
1410:
1342:
1267:law of sines
1204:
1159:
1143:
1098:
1084:
1014:
839:
719:
595:
529:
440:
185:
155:
143:
137:
118:
98:
70:circumcenter
63:
34:
28:
18:
3148:: 231–236,
2639:orthocenter
2600:orthocenter
2577:Nagel point
2371:orthocenter
2277:equilateral
840:Therefore,
182:Ratio proof
158:medial axis
74:orthocenter
51:medial axis
47:equidistant
3442:"Incenter"
3167:2014-10-28
2904:References
2627:Euler line
2621:Euler line
926:, so that
88:, and the
78:Euler line
3447:MathWorld
3392:218541138
3339:121434528
2983:Euclid's
2453:
2444:
2435:
2416:−
2326:−
2240:−
2109:⋅
2100:⋅
2053:⋅
2036:⋅
2012:⋅
1995:⋅
1971:⋅
1954:⋅
1319:
1301:
1283:
1055:∠
1033:¯
998:¯
980:¯
962:¯
944:¯
912:¯
894:¯
876:¯
858:¯
823:¯
805:¯
787:¯
769:¯
728:△
713:, by the
699:¯
681:¯
663:¯
645:¯
604:△
570:∠
548:¯
491:¯
459:¯
402:¯
370:¯
318:¯
277:∠
235:¯
194:∠
147:excircles
3460:Category
3239:Archived
3216:Archived
3117:(1980),
2985:Elements
2967:Archived
2635:centroid
2598:and the
2183:centroid
2166:inradius
505:meet at
441:And let
416:meet at
332:meet at
249:meet at
138:Elements
119:It is a
101:polygons
66:centroid
35:incenter
31:geometry
3331:2430410
3154:2877263
3072:1392429
2946:1573021
2938:2690608
2853:. Then
2579:of the
2286:of the
121:theorem
109:tangent
92:of the
3390:
3337:
3329:
3291:
3152:
3070:
2944:
2936:
2637:, and
2279:case.
2267:where
2187:median
2152:where
1594:, and
1508:, and
1383:, and
1343:where
1245:, and
1205:where
1175:
1114:
352:, and
177:Proofs
133:Euclid
72:, and
33:, the
3388:S2CID
3335:S2CID
3161:(PDF)
3138:(PDF)
3025:(PDF)
2934:JSTOR
2605:(the
1269:) by
1146:group
3289:ISBN
2845:Let
2800:<
2758:<
2719:<
2706:<
2693:<
2625:The
2592:disk
2551:<
2529:<
2510:<
2491:<
2338:<
2271:and
2164:and
2156:and
1411:The
1160:The
1099:The
1015:So
473:and
384:and
300:and
217:and
156:The
99:For
3380:doi
3319:doi
3097:doi
3060:doi
2926:doi
2450:cos
2441:cos
2432:cos
2376:is
2290:is
2192:By
1930:as
1928:ABC
1316:sin
1298:sin
1280:sin
720:In
596:In
135:'s
123:in
84:'s
29:In
3462::
3444:.
3386:,
3376:83
3374:,
3341:,
3333:,
3327:MR
3325:,
3315:52
3313:,
3188:^
3150:MR
3146:11
3144:,
3140:,
3093:96
3091:,
3068:MR
3066:,
3054:,
3050:,
2988:,
2954:^
2942:MR
2940:,
2932:,
2922:67
2920:,
2857:=
2633:,
2587:.
2068:1.
1574:,
1462:,
1363:,
1225:,
1152:.
1129:1.
1012:.
837:.
751:,
717:.
627:,
593:.
527:.
438:.
153:.
68:,
3450:.
3395:.
3382::
3346:.
3321::
3104:.
3099::
3075:.
3062::
3056:1
3032:.
3008:.
2949:.
2928::
2884:X
2881:C
2876:X
2873:B
2859:I
2855:X
2851:A
2847:X
2816:.
2813:R
2808:2
2805:1
2797:d
2774:;
2771:e
2766:3
2763:1
2755:d
2732:;
2727:3
2724:1
2714:v
2711:d
2701:u
2698:d
2688:s
2685:d
2670:s
2666:e
2662:R
2658:u
2654:v
2650:d
2615:G
2603:H
2596:G
2560:.
2557:O
2554:I
2548:N
2545:I
2542:2
2538:,
2535:O
2532:I
2526:G
2523:I
2519:,
2516:G
2513:H
2507:H
2504:I
2500:,
2497:G
2494:H
2488:G
2485:I
2459:.
2456:C
2447:B
2438:A
2427:2
2423:R
2419:4
2411:2
2407:r
2403:2
2400:=
2395:2
2391:H
2387:I
2374:H
2354:.
2351:R
2346:2
2343:1
2335:)
2332:r
2329:2
2323:R
2320:(
2315:2
2312:1
2307:=
2304:N
2301:I
2284:N
2273:r
2269:R
2252:,
2249:)
2246:r
2243:2
2237:R
2234:(
2231:R
2228:=
2223:2
2219:I
2215:O
2202:O
2198:I
2158:r
2154:R
2137:,
2132:2
2128:r
2124:R
2121:4
2118:=
2115:C
2112:I
2106:B
2103:I
2097:A
2094:I
2065:=
2059:A
2056:C
2050:C
2047:B
2042:C
2039:I
2033:C
2030:I
2024:+
2018:C
2015:B
2009:B
2006:A
2001:B
1998:I
1992:B
1989:I
1983:+
1977:B
1974:A
1968:A
1965:C
1960:A
1957:I
1951:A
1948:I
1932:I
1906:.
1900:c
1897:+
1894:b
1891:+
1888:a
1883:)
1878:C
1874:y
1870:,
1865:C
1861:x
1857:(
1854:c
1851:+
1848:)
1843:B
1839:y
1835:,
1830:B
1826:x
1822:(
1819:b
1816:+
1813:)
1808:A
1804:y
1800:,
1795:A
1791:x
1787:(
1784:a
1778:=
1773:)
1765:c
1762:+
1759:b
1756:+
1753:a
1746:C
1742:y
1738:c
1735:+
1730:B
1726:y
1722:b
1719:+
1714:A
1710:y
1706:a
1700:,
1694:c
1691:+
1688:b
1685:+
1682:a
1675:C
1671:x
1667:c
1664:+
1659:B
1655:x
1651:b
1648:+
1643:A
1639:x
1635:a
1627:(
1602:c
1582:b
1562:a
1542:)
1537:C
1533:y
1529:,
1524:C
1520:x
1516:(
1496:)
1491:B
1487:y
1483:,
1478:B
1474:x
1470:(
1450:)
1445:A
1441:y
1437:,
1432:A
1428:x
1424:(
1391:C
1371:B
1351:A
1328:)
1325:C
1322:(
1313::
1310:)
1307:B
1304:(
1295::
1292:)
1289:A
1286:(
1253:c
1233:b
1213:a
1190:c
1187::
1184:b
1181::
1178:a
1126::
1123:1
1120::
1117:1
1065:B
1062:C
1059:A
1029:F
1026:C
994:F
991:B
985::
976:F
973:A
967:=
958:C
955:B
949::
940:C
937:A
908:F
905:B
899::
890:C
887:B
881:=
872:F
869:A
863::
854:C
851:A
819:F
816:I
810::
801:I
798:C
792:=
783:F
780:B
774::
765:C
762:B
738:F
735:C
732:B
695:F
692:I
686::
677:I
674:C
668:=
659:F
656:A
650::
641:C
638:A
614:F
611:C
608:A
580:B
577:C
574:A
544:I
541:C
514:F
487:B
484:A
455:I
452:C
425:I
398:E
395:B
366:D
363:A
340:E
314:C
311:A
287:C
284:B
281:A
257:D
231:C
228:B
204:C
201:A
198:B
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
