170:. These are circles that are each tangent to the three lines through the triangle's sides. Each excircle touches one of these lines from the opposite side of the triangle, and is on the same side as the triangle for the other two lines. Like the incircle, the excircles are all tangent to the nine-point circle. Their points of tangency with the nine-point circle form a triangle, the Feuerbach triangle.
20:
1112:(1866), "On the Equations and Properties: (1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane",
507:
The latter property also holds for the tangency point of any of the excircles with the nine–point circle: the greatest distance from this tangency to one of the original triangle's side midpoints equals the sum of the distances to the other two side midpoints.
845:
The three lines from the vertices of the original triangle through the corresponding vertices of the
Feuerbach triangle meet at another triangle center, listed as X(12) in the Encyclopedia of Triangle Centers. Its trilinear coordinates are:
486:
826:
954:
679:
327:
144:
is another circle defined from a triangle. It is so called because it passes through nine significant points of the triangle, among which the simplest to construct are the
1050:
Eigenschaften einiger merkwürdigen Punkte des geradlinigen
Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung
1372:
335:
247:
227:
207:
1027:
694:
76:
1237:
852:
685:
577:
178:
The
Feuerbach point lies on the line through the centers of the two tangent circles that define it. These centers are the
1109:
102:
71:, meaning that its definition does not depend on the placement and scale of the triangle. It is listed as X(11) in
1367:
106:
137:
of the triangle, lies at the point where the three internal angle bisectors of the triangle cross each other.
1337:
Nguyen, Minh Ha; Nguyen, Pham Dat (2012), "Synthetic proofs of two theorems related to the
Feuerbach point",
332:
or, equivalently, the largest of the three distances equals the sum of the other two. Specifically, we have
89:, published by Feuerbach in 1822, states more generally that the nine-point circle is tangent to the three
1044:
80:
267:
568:
1139:
Chou, Shang-Ching (1988), "An introduction to Wu's method for mechanical theorem proving in geometry",
148:
of the triangle's sides. The nine-point circle passes through these three midpoints; thus, it is the
1254:
1164:
1121:
1081:
992:
94:
1183:
183:
141:
56:
24:
1246:
1148:
984:
72:
1350:
1329:
1308:
1287:
1266:
1160:
1095:
1004:
1346:
1325:
1304:
1283:
1262:
1156:
1091:
1048:
1031:
1024:
1000:
481:{\displaystyle x={\frac {R}{2OI}}|b-c|,\,y={\frac {R}{2OI}}|c-a|,z={\frac {R}{2OI}}|a-b|,}
250:
160:
153:
68:
60:
28:
1215:
Sa ́ndor
Nagydobai Kiss, "A Distance Property of the Feuerbach Point and Its Extension",
975:
Kimberling, Clark (1994), "Central Points and
Central Lines in the Plane of a Triangle",
1220:
1066:
232:
212:
192:
1361:
836:
93:
of the triangle as well as its incircle. A very short proof of this theorem based on
1186:
1168:
988:
493:
149:
163:
to each other. That point of tangency is the
Feuerbach point of the triangle.
1274:
Emelyanov, Lev; Emelyanova, Tatiana (2001), "A note on the
Feuerbach point",
1191:
167:
98:
36:
1316:
Vonk, Jan (2009), "The
Feuerbach point and reflections of the Euler line",
1295:
Suceavă, Bogdan; Yiu, Paul (2006), "The
Feuerbach point and Euler lines",
501:
179:
145:
134:
122:
90:
52:
48:
44:
32:
1125:
16:
Point where the incircle and nine-point circle of a triangle are tangent
1258:
1152:
996:
166:
Associated with the incircle of a triangle are three more circles, the
19:
249:
be the three distances of the Feuerbach point to the vertices of the
130:
1250:
133:
that is tangent to all three sides of the triangle. Its center, the
105:
in 1866; Feuerbach's theorem has also been used as a test case for
1086:
527:
respectively, and the midpoints of these sides are respectively
821:{\displaystyle (s-a)(b-c)^{2}:(s-b)(c-a)^{2}:(s-c)(a-b)^{2},}
39:
of a triangle. The incircle tangency is the Feuerbach point.
109:. The three points of tangency with the excircles form the
101:
of four circles tangent to a fifth circle was published by
159:
These two circles meet in a single point, where they are
855:
697:
580:
338:
270:
235:
215:
195:
1235:
Thébault, Victor (1949), "On the Feuerbach points",
1221:
http://forumgeom.fau.edu/FG2016volume16/FG201634.pdf
949:{\displaystyle 1+\cos(B-C):1+\cos(C-A):1+\cos(A-B).}
674:{\displaystyle 1-\cos(B-C):1-\cos(C-A):1-\cos(A-B).}
948:
820:
673:
511:If the incircle of triangle ABC touches the sides
480:
321:
241:
221:
201:
292:
1129:. See in particular the bottom of p. 411.
1067:"A simple vector proof of Feuerbach's theorem"
261:respectively of the original triangle). Then,
8:
1047:; Buzengeiger, Carl Heribert Ignatz (1822),
67:of the triangle. The Feuerbach point is a
1085:
854:
809:
769:
729:
696:
579:
470:
456:
438:
424:
410:
392:
385:
377:
363:
345:
337:
269:
234:
214:
194:
1053:(Monograph ed.), Nürnberg: Wiessner
18:
964:
1211:
1209:
1207:
1205:
1203:
1114:Proceedings of the Royal Irish Academy
1020:
1018:
1016:
1014:
970:
968:
7:
1373:Theorems about triangles and circles
322:{\displaystyle x+y+z=2\max(x,y,z),}
14:
1025:Encyclopedia of Triangle Centers
77:Encyclopedia of Triangle Centers
1141:Journal of Automated Reasoning
1065:Scheer, Michael J. G. (2011),
989:10.1080/0025570X.1994.11996210
940:
928:
910:
898:
880:
868:
806:
793:
790:
778:
766:
753:
750:
738:
726:
713:
710:
698:
665:
653:
635:
623:
605:
593:
471:
457:
425:
411:
378:
364:
313:
295:
1:
1238:American Mathematical Monthly
555:are similar to the triangles
59:of a triangle are internally
571:for the Feuerbach point are
539:, then with Feuerbach point
492:is the reference triangle's
253:(the midpoints of the sides
1389:
107:automated theorem proving
23:Feuerbach's theorem: the
1282:: 121–124 (electronic),
1045:Feuerbach, Karl Wilhelm
1030:April 19, 2012, at the
686:barycentric coordinates
113:of the given triangle.
1034:, accessed 2014-10-24.
950:
822:
675:
482:
323:
243:
223:
203:
81:Karl Wilhelm Feuerbach
40:
951:
823:
676:
569:trilinear coordinates
483:
324:
244:
224:
204:
79:, and is named after
63:to each other at the
22:
977:Mathematics Magazine
853:
695:
578:
336:
268:
233:
213:
193:
1339:Forum Geometricorum
1318:Forum Geometricorum
1297:Forum Geometricorum
1276:Forum Geometricorum
1219:16, 2016, 283–290.
1217:Forum Geometricorum
1074:Forum Geometricorum
87:Feuerbach's theorem
1184:Weisstein, Eric W.
1153:10.1007/BF00244942
946:
835:is the triangle's
818:
671:
478:
319:
239:
219:
199:
111:Feuerbach triangle
41:
1187:"Feuerbach Point"
454:
408:
361:
242:{\displaystyle z}
222:{\displaystyle y}
202:{\displaystyle x}
186:of the triangle.
184:nine-point center
142:nine-point circle
57:nine-point circle
25:nine-point circle
1380:
1368:Triangle centers
1353:
1332:
1311:
1290:
1269:
1223:
1213:
1198:
1197:
1196:
1179:
1173:
1171:
1136:
1130:
1128:
1106:
1100:
1098:
1089:
1071:
1062:
1056:
1054:
1041:
1035:
1022:
1009:
1007:
972:
955:
953:
952:
947:
827:
825:
824:
819:
814:
813:
774:
773:
734:
733:
680:
678:
677:
672:
487:
485:
484:
479:
474:
460:
455:
453:
439:
428:
414:
409:
407:
393:
381:
367:
362:
360:
346:
328:
326:
325:
320:
248:
246:
245:
240:
228:
226:
225:
220:
208:
206:
205:
200:
73:Clark Kimberling
1388:
1387:
1383:
1382:
1381:
1379:
1378:
1377:
1358:
1357:
1336:
1315:
1294:
1273:
1251:10.2307/2305531
1234:
1231:
1229:Further reading
1226:
1214:
1201:
1182:
1181:
1180:
1176:
1138:
1137:
1133:
1108:
1107:
1103:
1069:
1064:
1063:
1059:
1043:
1042:
1038:
1032:Wayback Machine
1023:
1012:
974:
973:
966:
962:
851:
850:
805:
765:
725:
693:
692:
576:
575:
565:
443:
397:
350:
334:
333:
266:
265:
251:medial triangle
231:
230:
211:
210:
191:
190:
176:
154:medial triangle
119:
95:Casey's theorem
69:triangle center
65:Feuerbach point
17:
12:
11:
5:
1386:
1384:
1376:
1375:
1370:
1360:
1359:
1356:
1355:
1334:
1313:
1292:
1271:
1245:(8): 546–547,
1230:
1227:
1225:
1224:
1199:
1174:
1147:(3): 237–267,
1131:
1101:
1057:
1036:
1010:
983:(3): 163–187,
963:
961:
958:
957:
956:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
829:
828:
817:
812:
808:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
772:
768:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
732:
728:
724:
721:
718:
715:
712:
709:
706:
703:
700:
682:
681:
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:
564:
561:
559:respectively.
543:the triangles
477:
473:
469:
466:
463:
459:
452:
449:
446:
442:
437:
434:
431:
427:
423:
420:
417:
413:
406:
403:
400:
396:
391:
388:
384:
380:
376:
373:
370:
366:
359:
356:
353:
349:
344:
341:
330:
329:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
238:
218:
198:
175:
172:
125:of a triangle
118:
115:
15:
13:
10:
9:
6:
4:
3:
2:
1385:
1374:
1371:
1369:
1366:
1365:
1363:
1352:
1348:
1344:
1340:
1335:
1331:
1327:
1323:
1319:
1314:
1310:
1306:
1302:
1298:
1293:
1289:
1285:
1281:
1277:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1239:
1233:
1232:
1228:
1222:
1218:
1212:
1210:
1208:
1206:
1204:
1200:
1194:
1193:
1188:
1185:
1178:
1175:
1170:
1166:
1162:
1158:
1154:
1150:
1146:
1142:
1135:
1132:
1127:
1123:
1119:
1115:
1111:
1105:
1102:
1097:
1093:
1088:
1083:
1079:
1075:
1068:
1061:
1058:
1052:
1051:
1046:
1040:
1037:
1033:
1029:
1026:
1021:
1019:
1017:
1015:
1011:
1006:
1002:
998:
994:
990:
986:
982:
978:
971:
969:
965:
959:
943:
937:
934:
931:
925:
922:
919:
916:
913:
907:
904:
901:
895:
892:
889:
886:
883:
877:
874:
871:
865:
862:
859:
856:
849:
848:
847:
843:
842:
838:
837:semiperimeter
834:
815:
810:
802:
799:
796:
787:
784:
781:
775:
770:
762:
759:
756:
747:
744:
741:
735:
730:
722:
719:
716:
707:
704:
701:
691:
690:
689:
687:
668:
662:
659:
656:
650:
647:
644:
641:
638:
632:
629:
626:
620:
617:
614:
611:
608:
602:
599:
596:
590:
587:
584:
581:
574:
573:
572:
570:
562:
560:
558:
557:AOI, BOI, COI
554:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
509:
505:
503:
499:
495:
491:
475:
467:
464:
461:
450:
447:
444:
440:
435:
432:
429:
421:
418:
415:
404:
401:
398:
394:
389:
386:
382:
374:
371:
368:
357:
354:
351:
347:
342:
339:
316:
310:
307:
304:
301:
298:
289:
286:
283:
280:
277:
274:
271:
264:
263:
262:
260:
256:
252:
236:
216:
196:
187:
185:
181:
173:
171:
169:
164:
162:
157:
155:
151:
147:
143:
138:
136:
132:
128:
124:
116:
114:
112:
108:
104:
100:
96:
92:
88:
84:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
38:
34:
30:
26:
21:
1342:
1338:
1321:
1317:
1300:
1296:
1279:
1275:
1242:
1236:
1216:
1190:
1177:
1144:
1140:
1134:
1117:
1113:
1104:
1077:
1073:
1060:
1049:
1039:
980:
976:
844:
840:
832:
830:
683:
566:
556:
552:
548:
544:
540:
536:
532:
528:
524:
520:
516:
512:
510:
506:
497:
494:circumcenter
489:
331:
258:
254:
188:
177:
165:
158:
150:circumcircle
139:
126:
120:
117:Construction
110:
86:
85:
64:
42:
1303:: 191–197,
1120:: 396–423,
1080:: 205–210,
563:Coordinates
1362:Categories
960:References
513:BC, CA, AB
255:BC=a, CA=b
174:Properties
103:John Casey
99:bitangents
1345:: 39–46,
1324:: 47–55,
1192:MathWorld
1110:Casey, J.
1087:1107.1152
935:−
926:
905:−
896:
875:−
866:
841:a+b+c)/2.
800:−
785:−
760:−
745:−
720:−
705:−
660:−
651:
645:−
630:−
621:
615:−
600:−
591:
585:−
465:−
419:−
372:−
168:excircles
146:midpoints
91:excircles
49:triangles
37:excircles
1169:12368370
1126:20488927
1028:Archived
502:incenter
180:incenter
135:incenter
123:incircle
53:incircle
45:geometry
33:incircle
1351:2955643
1330:2534378
1309:2282236
1288:1891524
1267:0033039
1259:2305531
1161:0975146
1096:2877268
1005:1573021
997:2690608
551:, and
500:is its
488:where
161:tangent
152:of the
97:on the
61:tangent
43:In the
31:to the
29:tangent
1349:
1328:
1307:
1286:
1265:
1257:
1167:
1159:
1124:
1094:
1003:
995:
831:where
535:, and
523:, and
257:, and
229:, and
131:circle
51:, the
1255:JSTOR
1165:S2CID
1122:JSTOR
1082:arXiv
1070:(PDF)
993:JSTOR
129:is a
688:are
684:Its
567:The
496:and
259:AB=c
189:Let
182:and
140:The
121:The
75:'s
55:and
35:and
1247:doi
1149:doi
985:doi
923:cos
893:cos
863:cos
648:cos
618:cos
588:cos
553:FRZ
549:FQY
545:FPX
515:at
293:max
127:ABC
47:of
27:is
1364::
1347:MR
1343:12
1341:,
1326:MR
1320:,
1305:MR
1299:,
1284:MR
1278:,
1263:MR
1261:,
1253:,
1243:56
1241:,
1202:^
1189:.
1163:,
1157:MR
1155:,
1143:,
1116:,
1092:MR
1090:,
1078:11
1076:,
1072:,
1013:^
1001:MR
999:,
991:,
981:67
979:,
967:^
547:,
531:,
519:,
504:.
209:,
156:.
83:.
1354:.
1333:.
1322:9
1312:.
1301:6
1291:.
1280:1
1270:.
1249::
1195:.
1172:.
1151::
1145:4
1118:9
1099:.
1084::
1055:.
1008:.
987::
944:.
941:)
938:B
932:A
929:(
920:+
917:1
914::
911:)
908:A
902:C
899:(
890:+
887:1
884::
881:)
878:C
872:B
869:(
860:+
857:1
839:(
833:s
816:,
811:2
807:)
803:b
797:a
794:(
791:)
788:c
782:s
779:(
776::
771:2
767:)
763:a
757:c
754:(
751:)
748:b
742:s
739:(
736::
731:2
727:)
723:c
717:b
714:(
711:)
708:a
702:s
699:(
669:.
666:)
663:B
657:A
654:(
642:1
639::
636:)
633:A
627:C
624:(
612:1
609::
606:)
603:C
597:B
594:(
582:1
541:F
537:R
533:Q
529:P
525:Z
521:Y
517:X
498:I
490:O
476:,
472:|
468:b
462:a
458:|
451:I
448:O
445:2
441:R
436:=
433:z
430:,
426:|
422:a
416:c
412:|
405:I
402:O
399:2
395:R
390:=
387:y
383:,
379:|
375:c
369:b
365:|
358:I
355:O
352:2
348:R
343:=
340:x
317:,
314:)
311:z
308:,
305:y
302:,
299:x
296:(
290:2
287:=
284:z
281:+
278:y
275:+
272:x
237:z
217:y
197:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
