112:
712:
452:
707:{\displaystyle {\begin{aligned}\angle DIA&=180^{\circ }-\angle AIB\\&=180^{\circ }-(180^{\circ }-\angle IAB-\angle IBA)\\&=\angle IAB+\angle IBA\\&=\angle IAC+\angle DAC\\&=\angle IAD\\\implies AD&=DI.\end{aligned}}}
102:
is the circumcircle of the original triangle. The theorem is helpful for solving competitive
Euclidean geometry problems, and can be used to reconstruct a triangle starting from one vertex, the incenter, and the circumcenter.
350:
442:
457:
375:
1375:
807:
is tangent to the circumcircle or because the two circles do not have two crossing points. It may also produce a triangle for which the given point
1353:
1289:
1132:
930:
275:
1069:
989:
826:
Other triangle reconstruction problems, such as the reconstruction of a triangle from a vertex, incenter, and center of its
380:
1051:
1222:
1086:
Garcia, Ronaldo; Odehnal, Boris; Reznik, Dan (2022). "Loci of poncelet triangles in the general closure case".
1171:
899:
36:
722:
This theorem can be used to reconstruct a triangle starting from the locations only of one vertex, the
162:
1150:
1252:
1248:
1218:
95:
1188:
1095:
981:
977:
225:
878:
as the (internal) angle bisector of one of the triangle's angles. However, it is also true when
1349:
1343:
1285:
1128:
1045:
985:
926:
827:
99:
56:
1279:
1007:
1003:
830:, can be solved by reducing the problem to the case of a vertex, incenter, and circumcenter.
1180:
1149:
1105:
1329:
1325:
1022:
846:
be any two of the four points given by the incenter and the three excenters of a triangle
269:
862:
as diameter passes through the other two vertices and is centered on the circumcircle of
811:
is an excenter rather than the incenter. In these cases, there can be no triangle having
357:
151:
209:. In particular, this implies that the center of this circle lies on the circumcircle.
1369:
1311:"Conic construction of a triangle from its incenter, nine-point center, and a vertex"
1075:(in Russian). СУНЦ МГУ им. М. В. Ломоносова - школа им. А.Н. Колмогорова. 2014-10-29.
94:
These relationships arise because the incenter and excenters of any triangle form an
1034:
1257:
1227:
727:
60:
32:
174:
111:
1109:
742:
be the circumcenter. This information allows the successive construction of:
998:
1169:
Morris, Richard (1928), "Circles through notable points of the triangle",
1184:
723:
241:
139:
131:
84:
40:
20:
1192:
1310:
858:
are collinear with one of the three triangle vertices. The circle with
88:
28:
44:
1100:
16:
A statement about properties of inscribed and circumscribed circles
1028:(in Russian). Ф7 (Теорема трилистника), page 34; proof on page 36.
746:
the circumcircle of the given triangle, as the circle with center
110:
1348:. World Scientific. Examples 6.145 and 6.146, pp. 328–329.
890:
is the external angle bisector of one of the triangle's angles.
950:
63:. This theorem is best known in Russia, where it is called the
1033:Р. Н. Карасёв; В. Л. Дольников; И. И. Богданов; А. В. Акопян.
345:{\displaystyle \angle IBA=\angle DCA,\ \angle IBC=\angle DAC.}
1342:
Chou, Shang-Ching; Gao, Xiao-Shan; Zhang, Jingzhong (1994).
1196:. See in particular the discussion on p. 65 of circles
91:; these names have sometimes also been adopted in English.
874:
is the incenter, this is the trillium theorem, with line
921:
Chen, Evan (2016). "§1.4 The
Incenter/Excenter Lemma".
951:"Incenter Symmetry, Euler Lines, and Schiffler Points"
925:. Mathematical Association of America. pp. 9–10.
455:
383:
360:
278:
1123:
Zaslavsky, Alexey A.; Skopenkov, Mikhail B. (2021).
437:{\displaystyle \angle DCA=\angle DAC\implies AD=CD.}
83:), based on the geometric figure's resemblance to a
1253:"Midpoints of the Lines Joining In- and Excenters"
803:, this construction may fail, either because line
761:as the intersection of the circumcircle with line
706:
436:
369:
344:
39:of a triangle, or between two excenters, is the
787:as the intersection points of the two circles.
1127:. American Mathematical Society. p. 15.
1035:"Задачи для школьного математического кружка"
1023:"Это открытие - золотой ключ Леонарда Эйлера"
78:
68:
8:
1281:Problems and Solutions in Euclidean Geometry
923:Euclidean Geometry in Mathematical Olympiads
1223:"A Property of Circle Through the Incenter"
1125:Mathematics via Problems. Part 2: Geometry
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677:
673:
415:
411:
1099:
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518:
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456:
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359:
277:
974:CRC Concise Encyclopedia of Mathematics
949:Le, Nguyen; Wildberger, Norman (2016).
910:
886:are both excenters; in this case, line
768:the circle of the theorem, with center
1304:
1302:
1278:Aref, M. N.; Wernick, William (1968).
1043:
916:
914:
718:Application to triangle reconstruction
252:, also lies at the same distance from
115:incenter–excenter lemma with incenter
1158:. Houghton Mifflin. pp. 182–194.
1040:(in Russian). Problem 1.2. p. 4.
7:
1376:Theorems about triangles and circles
944:
942:
791:However, for some triples of points
55:) also passing through two triangle
1151:"X. Inscribed and Escribed Circles"
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14:
1318:Journal for Geometry and Graphics
169:. Then, the theorem states that
674:
570:
527:
412:
256:, diametrically opposite from
1:
1284:. Dover. 3.3(i), p. 68.
972:Weisstein, Eric W. (1999).
1392:
1345:Machine Proofs in Geometry
1148:Johnson, Roger A. (1929).
1110:10.1007/s00022-022-00629-3
1008:"Incenter–Excenter Circle"
1004:"Excenter–Excenter Circle"
982:"Incenter–Excenter Circle"
978:"Excenter–Excenter Circle"
730:of the triangle. For, let
79:
69:
37:incenter and any excenter
1050:: CS1 maint: location (
146:be the point where line
53:excenter–excenter circle
1172:The Mathematics Teacher
270:inscribed angle theorem
59:with its center on the
25:incenter–excenter lemma
900:Angle bisector theorem
708:
438:
377:is an angle bisector,
371:
346:
123:
1070:"6. Лемма о трезубце"
738:be the incenter, and
734:be the given vertex,
709:
439:
372:
347:
114:
1249:Bogomolny, Alexander
1219:Bogomolny, Alexander
1208:, and their centers.
1185:10.5951/MT.21.2.0069
453:
381:
358:
276:
212:The three triangles
1088:Journal of Geometry
193:The circle through
96:orthocentric system
70:теорема трилистника
1309:Yiu, Paul (2012),
1019:Trillium theorem:
704:
702:
434:
370:{\displaystyle BI}
367:
342:
205:has its center at
124:
828:nine-point circle
823:as circumcenter.
819:as incenter, and
311:
100:nine-point circle
49:incenter–excenter
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189:. Equivalently:
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130:be an arbitrary
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82:
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80:лемма о трезубце
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65:trillium theorem
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1156:Modern Geometry
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1031:Trident lemma:
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996:Republished at
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236:A fourth point
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1324:(2): 171–183,
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1021:И. А. Кушнир.
1012:
990:
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909:
907:
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903:
902:
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892:
866:. When one of
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834:Generalization
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232:as their apex.
210:
161:) crosses the
152:angle bisector
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15:
13:
10:
9:
6:
4:
3:
2:
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1355:9789810215842
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1291:9780486477206
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1134:9781470448790
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1024:
1016:
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1005:
1001:
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993:
987:
983:
979:
976:. CRC Press.
975:
968:
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952:
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932:9780883858394
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153:
141:
133:
119:and excenter
113:
106:
104:
101:
97:
92:
90:
86:
76:
75:trident lemma
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1344:
1337:
1321:
1317:
1280:
1273:
1262:. Retrieved
1258:Cut-the-Knot
1256:
1243:
1232:. Retrieved
1228:Cut-the-Knot
1226:
1213:
1179:(2): 63–71,
1176:
1170:
1164:
1155:
1143:
1124:
1118:
1091:
1087:
1081:
1030:
1015:
997:
973:
967:
961:(20): 22–30.
958:
954:
922:
837:
825:
790:
728:circumcenter
721:
446:We also get
445:
353:
267:
248:relative to
235:
163:circumcircle
157:
125:
93:
74:
64:
61:circumcircle
52:
48:
35:between the
33:line segment
24:
18:
815:as vertex,
772:and radius
750:and radius
175:equidistant
1264:2016-01-26
1234:2016-01-26
1101:2108.05430
991:0849396409
906:References
726:, and the
87:flower or
1094:(1): 17.
999:MathWorld
779:vertices
675:⟹
658:∠
636:∠
621:∠
599:∠
584:∠
559:∠
556:−
544:∠
541:−
536:∘
525:−
520:∘
493:∠
490:−
485:∘
461:∠
413:⟹
400:∠
385:∠
328:∠
313:∠
295:∠
280:∠
226:isosceles
107:Statement
31:that the
1370:Category
1193:27951001
1046:cite web
984:p. 894.
980:p. 591,
894:See also
724:incenter
242:excenter
142:and let
140:incenter
132:triangle
85:trillium
57:vertices
41:diameter
21:geometry
1330:3088369
850:. Then
268:By the
228:, with
138:be its
134:. Let
89:trident
29:theorem
27:is the
1352:
1328:
1288:
1191:
1131:
988:
929:
799:, and
757:point
354:Since
310:
240:, the
220:, and
201:, and
185:, and
98:whose
45:circle
23:, the
1314:(PDF)
1189:JSTOR
1096:arXiv
1073:(PDF)
1038:(PDF)
1026:(PDF)
776:, and
264:Proof
177:from
150:(the
73:) or
43:of a
1350:ISBN
1286:ISBN
1129:ISBN
1052:link
986:ISBN
927:ISBN
882:and
854:and
842:and
838:Let
783:and
224:are
126:Let
47:(an
1206:AIB
1202:CIA
1198:BIC
1181:doi
1106:doi
1092:113
955:KoG
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516:180
481:180
246:ABC
244:of
222:ACD
218:CID
214:AID
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167:ABC
165:of
158:ABC
154:of
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1301:^
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1358:.
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785:C
781:A
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740:O
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698:.
695:I
692:D
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655:=
645:C
642:A
639:D
633:+
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624:I
618:=
608:A
605:B
602:I
596:+
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587:I
581:=
571:)
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565:B
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464:D
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298:D
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289:A
286:B
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258:I
254:D
250:B
238:E
230:D
207:D
203:I
199:C
195:A
187:I
183:C
179:A
171:D
156:∠
144:D
136:I
121:E
117:I
77:(
67:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
