654:
31:
683:, and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines passes through three of the ten points, and each of the ten points lies on three of the ten lines. Those ten points and ten lines make up the
707:
This restricted version states that if two triangles are perspective from a point on a given line, and two pairs of corresponding sides also meet on this line, then the third pair of corresponding sides meet on the line as well. Thus, it is the specialization of
Desargues's Theorem to only the cases
524:
This proves
Desargues's theorem if the two triangles are not contained in the same plane. If they are in the same plane, Desargues's theorem can be proved by choosing a point not in the plane, using this to lift the triangles out of the plane so that the argument above works, and then projecting back
34:
Perspective triangles. Corresponding sides of the triangles, when extended, meet at points on a line called the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues's theorem states that the truth of
314:(where points correspond to lines and collinearity of points corresponds to concurrency of lines), the statement of Desargues's theorem is self-dual: axial perspectivity is translated into central perspectivity and vice versa. The Desargues configuration (below) is a self-dual configuration.
217:
but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the
Euclidean plane by adding points at infinity, following
317:
This self-duality in the statement is due to the usual modern way of writing the theorem. Historically, the theorem only read, "In a projective space, a pair of centrally perspective triangles is axially perspective" and the dual of this statement was called the
273:
The importance of
Desargues's theorem in abstract projective geometry is due especially to the fact that a projective space satisfies that theorem if and only if it is isomorphic to a projective space defined over a field or division ring.
290:
a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the simplest geometric theorems whose natural home is in projective rather than affine space.
603:
lie on a second line, then each two opposite sides of the hexagon lie on two lines that meet in a point and the three points constructed in this way are collinear. A plane in which Pappus's theorem is universally true is called
330:
Desargues's theorem holds for projective space of any dimension over any field or division ring, and also holds for abstract projective spaces of dimension at least 3. In dimension 2 the planes for which it holds are called
831:, pg. 159, footnote 1), Hessenberg's original proof is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by
699:
of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.
531:
also asserts that three points lie on a line, and has a proof using the same idea of considering it in three rather than two dimensions and writing the line as an intersection of two planes.
543:
in which
Desargues's theorem is not true, some extra conditions need to be met in order to prove it. These conditions usually take the form of assuming the existence of sufficiently many
347:
Desargues's theorem is true for any projective space of dimension at least 3, and more generally for any projective space that can be embedded in a space of dimension at least 3.
882:
619:
of this result is not true, that is, not all
Desarguesian planes are Pappian. Satisfying Pappus's theorem universally is equivalent to having the underlying coordinate system be
521:
also exist and belong to the planes of both triangles. Since these two planes intersect in more than one point, their intersection is a line that contains all three points.
525:
into the plane. The last step of the proof fails if the projective space has dimension less than 3, as in this case it is not possible to find a point not in the plane.
657:
The
Desargues configuration viewed as a pair of mutually inscribed pentagons: each pentagon vertex lies on the line through one of the sides of the other pentagon.
303:
if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be
623:. A plane defined over a non-commutative division ring (a division ring that is not a field) would therefore be Desarguesian but not Pappian. However, due to
1157:
Pambuccian, Victor; Schacht, Celia (2019), "The axiomatic destiny of the theorems of Pappus and
Desargues", in Dani, S. G.; Papadopoulos, A. (eds.),
980:
887:
1319:
691:. Although Desargues's theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more
1304:
1300:
1166:
1027:
961:
872:
639:
give a proof that uses only "elementary" algebraic facts (rather than the full strength of
Wedderburn's little theorem).
265:) to a practical book on the use of perspective published in 1648. by his friend and pupil Abraham Bosse (1602–1676).
1334:
1329:
1231:
1213:
1191:
1148:
1126:
1108:
1090:
928:
540:
479:
belong to the same plane and must intersect. Further, if the two triangles lie on different planes, then the point
949:
937:
624:
1251:
311:
547:
of a certain type, which in turn leads to showing that the underlying algebraic coordinate system must be a
1324:
17:
1241:
1246:
559:
242:
1183:
335:
and are the same as the planes that can be given coordinates over a division ring. There are also many
1282:
635:
Desarguesian planes are
Pappian. There is no known completely geometric proof of this fact, although
688:
36:
1273:
1291:
920:
684:
648:
336:
300:
246:
64:
1038:
612:
showed that Desargues's theorem can be deduced from three applications of Pappus's theorem.
304:
230:
1009:
971:
219:
210:
8:
1078:
724:
234:
44:
27:
Two triangles are in perspective axially if and only if they are in perspective centrally
1202:
1175:
1063:
997:
904:
661:
The ten lines involved in Desargues's theorem (six sides of triangles, the three lines
528:
332:
172:
meet in a third point, and that these three points all lie on a common line called the
1264:
1227:
1209:
1187:
1162:
1144:
1137:
1122:
1104:
1086:
1067:
1055:
1023:
957:
924:
908:
868:
715:
is a projective plane in which the little Desargues theorem is valid for every line.
82:
1036:
Hessenberg, Gerhard (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen",
1047:
989:
896:
223:
52:
1017:
1005:
967:
862:
287:
214:
257:
Desargues never published this theorem, but it appeared in an appendix entitled
653:
241:; that includes any projective space of dimension greater than two or in which
70:
1313:
1074:
1059:
712:
548:
238:
120:
1286:
1277:
544:
447:
are coplanar (lie in the same plane) because of the assumed concurrency of
283:
900:
616:
319:
620:
708:
in which the center of perspectivity lies on the axis of perspectivity.
30:
1295:
1051:
1001:
1268:
420:
993:
692:
60:
263:
Manière universelle de M. Desargues pour practiquer la perspective
563:
322:
of Desargues's theorem and was always referred to by that name.
493:
belongs to both planes. By a symmetric argument, the points
233:
and for any projective space defined arithmetically from a
978:
Cronheim, Arno (1953), "A proof of Hessenberg's theorem",
883:"Completing Segre's proof of Wedderburn's little theorem"
1143:(2nd ed.), Reading, Mass.: Addison Wesley Longman,
259:
Universal Method of M. Desargues for Using Perspective
1201:
1136:
1073:
277:
1156:
1311:
981:Proceedings of the American Mathematical Society
1174:
880:
794:
793:The smallest examples of these can be found in
702:
636:
554:
325:
1239:
642:
350:Desargues's theorem can be stated as follows:
860:
807:
861:Albert, A. Adrian; Sandler, Reuben (2015) ,
1098:
1085:(2nd ed.), Chelsea, pp. 119–128,
888:Bulletin of the London Mathematical Society
864:An Introduction to Finite Projective Planes
811:
1035:
609:
342:
249:", in which Desargues's theorem is false.
1221:
1015:
828:
815:
339:where Desargues's theorem does not hold.
1139:A History of Mathematics:An Introduction
1116:
977:
832:
652:
534:
29:
948:
936:
844:
781:
768:
14:
1312:
376:are concurrent (meet at a point), then
202:are concurrent, at a point called the
1199:
914:
881:Bamberg, John; Penttila, Tim (2015),
743:
571:is drawn in such a way that vertices
1134:
917:Projective Geometry: An Introduction
755:
312:duality of plane projective geometry
229:Desargues's theorem is true for the
268:
24:
541:non-Desarguesian projective planes
156:meet in a second point, and lines
25:
1346:
1258:
1119:Introduction to Finite Geometries
1099:Hughes, Dan; Piper, Fred (1973),
299:By definition, two triangles are
1222:Stevenson, Frederick W. (1972),
245:holds. However, there are many "
1320:Theorems in projective geometry
950:Coxeter, Harold Scott MacDonald
631:division rings are fields, all
294:
278:Projective versus affine spaces
1240:Voitsekhovskii, M.I. (2001) ,
1161:, Springer, pp. 355–399,
838:
821:
800:
787:
774:
761:
749:
737:
13:
1:
1178:; Kirkpatrick, P. B. (1971),
854:
637:Bamberg & Penttila (2015)
1292:Proof of Desargues's theorem
1204:A Source Book in Mathematics
1200:Smith, David Eugene (1959),
1083:Geometry and the Imagination
703:The little Desargues theorem
555:Relation to Pappus's theorem
326:Proof of Desargues's theorem
101:, and those of the other by
39:for the truth of the second.
7:
1247:Encyclopedia of Mathematics
795:Room & Kirkpatrick 1971
718:
643:The Desargues configuration
625:Wedderburn's little theorem
587:lie on a line and vertices
180:means that the three lines
10:
1351:
1184:Cambridge University Press
646:
252:
1305:Dynamic Geometry Sketches
1117:Kárteszi, Ferenc (1976),
1016:Dembowski, Peter (1968),
808:Albert & Sandler 2015
73:they are in perspective
1335:Euclidean plane geometry
1330:Theorems about triangles
1135:Katz, Victor J. (1998),
1046:(2), Springer: 161–172,
954:Introduction to Geometry
730:
689:projective configuration
627:, which states that all
560:Pappus's hexagon theorem
37:necessary and sufficient
1180:Miniquaternion Geometry
956:(2nd ed.), Wiley,
921:Oxford University Press
812:Hughes & Piper 1973
685:Desargues configuration
649:Desargues configuration
463:. Therefore, the lines
343:Three-dimensional proof
337:non-Desarguesian planes
247:non-Desarguesian planes
204:center of perspectivity
140:meet in a point, lines
35:the first condition is
1242:"Desargues assumption"
847:, p. 238, section 14.3
658:
40:
1039:Mathematische Annalen
656:
535:Two-dimensional proof
231:real projective plane
213:is true in the usual
178:Central perspectivity
174:axis of perspectivity
33:
1079:Cohn-Vossen, Stephan
222:. This results in a
220:Jean-Victor Poncelet
211:intersection theorem
1325:Proof without words
1301:Desargues's Theorem
1283:Monge via Desargues
1274:Desargues's Theorem
1159:Geometry in history
1103:, Springer-Verlag,
1022:, Springer Verlag,
942:Projective Geometry
915:Casse, Rey (2006),
901:10.1112/blms/bdv021
333:Desarguesian planes
310:Under the standard
85:of one triangle by
49:Desargues's theorem
45:projective geometry
1052:10.1007/BF01457558
771:) pp. 26–27.
687:, an example of a
659:
562:states that, if a
41:
1265:Desargues Theorem
1224:Projective Planes
1168:978-3-030-13611-6
1121:, North-Holland,
1101:Projective Planes
1029:978-3-540-61786-0
1019:Finite Geometries
963:978-0-471-50458-0
874:978-0-486-78994-1
610:Hessenberg (1905)
124:means that lines
81:Denote the three
16:(Redirected from
1342:
1254:
1236:
1226:, W.H. Freeman,
1218:
1207:
1196:
1171:
1153:
1142:
1131:
1113:
1095:
1070:
1032:
1012:
974:
945:
933:
911:
877:
848:
842:
836:
825:
819:
804:
798:
791:
785:
778:
772:
765:
759:
753:
747:
741:
725:Pascal's theorem
682:
681:
674:
673:
667:
602:
596:
586:
580:
570:
520:
519:
513:
506:
505:
499:
492:
491:
485:
478:
477:
470:
469:
462:
461:
454:
453:
446:
440:
418:
417:
411:
404:
403:
397:
391:
385:
375:
374:
367:
366:
360:
269:Coordinatization
243:Pappus's theorem
224:projective plane
201:
200:
193:
192:
186:
171:
170:
163:
162:
155:
154:
147:
146:
139:
138:
131:
130:
116:
110:
100:
94:
53:Girard Desargues
21:
1350:
1349:
1345:
1344:
1343:
1341:
1340:
1339:
1310:
1309:
1261:
1234:
1216:
1194:
1176:Room, Thomas G.
1169:
1151:
1129:
1111:
1093:
1030:
994:10.2307/2031794
964:
938:Coxeter, H.S.M.
931:
875:
857:
852:
851:
843:
839:
826:
822:
805:
801:
792:
788:
779:
775:
766:
762:
754:
750:
742:
738:
733:
721:
705:
677:
676:
669:
663:
662:
651:
645:
598:
588:
582:
572:
566:
557:
537:
529:Monge's theorem
515:
509:
508:
501:
495:
494:
487:
481:
480:
473:
472:
465:
464:
457:
456:
449:
448:
442:
428:
413:
407:
406:
399:
393:
387:
381:
380:
370:
369:
362:
356:
355:
345:
328:
297:
288:Euclidean plane
280:
271:
255:
215:Euclidean plane
196:
195:
188:
182:
181:
166:
165:
158:
157:
150:
149:
142:
141:
134:
133:
126:
125:
112:
102:
96:
86:
28:
23:
22:
15:
12:
11:
5:
1348:
1338:
1337:
1332:
1327:
1322:
1308:
1307:
1298:
1289:
1280:
1271:
1260:
1259:External links
1257:
1256:
1255:
1237:
1232:
1219:
1214:
1197:
1192:
1172:
1167:
1154:
1149:
1132:
1127:
1114:
1109:
1096:
1091:
1075:Hilbert, David
1071:
1033:
1028:
1013:
988:(2): 219–221,
975:
962:
946:
934:
929:
912:
895:(3): 483–492,
878:
873:
856:
853:
850:
849:
837:
829:Dembowski 1968
827:According to (
820:
816:Stevenson 1972
799:
786:
773:
760:
748:
735:
734:
732:
729:
728:
727:
720:
717:
704:
701:
647:Main article:
644:
641:
556:
553:
536:
533:
425:
424:
377:
344:
341:
327:
324:
296:
293:
279:
276:
270:
267:
254:
251:
79:
78:
71:if and only if
51:, named after
26:
9:
6:
4:
3:
2:
1347:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1317:
1315:
1306:
1302:
1299:
1297:
1293:
1290:
1288:
1284:
1281:
1279:
1275:
1272:
1270:
1266:
1263:
1262:
1253:
1249:
1248:
1243:
1238:
1235:
1233:0-7167-0443-9
1229:
1225:
1220:
1217:
1215:0-486-64690-4
1211:
1206:
1205:
1198:
1195:
1193:0-521-07926-8
1189:
1185:
1182:, Cambridge:
1181:
1177:
1173:
1170:
1164:
1160:
1155:
1152:
1150:0-321-01618-1
1146:
1141:
1140:
1133:
1130:
1128:0-7204-2832-7
1124:
1120:
1115:
1112:
1110:0-387-90044-6
1106:
1102:
1097:
1094:
1092:0-8284-1087-9
1088:
1084:
1080:
1076:
1072:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1040:
1034:
1031:
1025:
1021:
1020:
1014:
1011:
1007:
1003:
999:
995:
991:
987:
983:
982:
976:
973:
969:
965:
959:
955:
951:
947:
943:
939:
935:
932:
930:0-19-929886-6
926:
922:
918:
913:
910:
906:
902:
898:
894:
890:
889:
884:
879:
876:
870:
866:
865:
859:
858:
846:
841:
834:
833:Cronheim 1953
830:
824:
817:
813:
809:
803:
796:
790:
783:
777:
770:
764:
757:
752:
745:
740:
736:
726:
723:
722:
716:
714:
713:Moufang plane
709:
700:
698:
694:
690:
686:
680:
672:
666:
655:
650:
640:
638:
634:
630:
626:
622:
618:
613:
611:
607:
601:
595:
591:
585:
579:
575:
569:
565:
561:
552:
551:(skewfield).
550:
549:division ring
546:
545:collineations
542:
539:As there are
532:
530:
526:
522:
518:
512:
504:
498:
490:
484:
476:
468:
460:
452:
445:
439:
435:
431:
422:
416:
410:
402:
396:
390:
384:
378:
373:
365:
359:
353:
352:
351:
348:
340:
338:
334:
323:
321:
315:
313:
308:
306:
302:
292:
289:
285:
275:
266:
264:
260:
250:
248:
244:
240:
239:division ring
236:
232:
227:
225:
221:
216:
212:
207:
205:
199:
191:
185:
179:
175:
169:
161:
153:
145:
137:
129:
123:
122:
121:perspectivity
115:
109:
105:
99:
93:
89:
84:
76:
72:
69:
66:
62:
58:
57:
56:
54:
50:
46:
38:
32:
19:
1287:cut-the-knot
1278:cut-the-knot
1245:
1223:
1203:
1179:
1158:
1138:
1118:
1100:
1082:
1043:
1037:
1018:
985:
979:
953:
941:
916:
892:
886:
863:
845:Coxeter 1969
840:
823:
802:
789:
782:Coxeter 1964
776:
769:Coxeter 1964
763:
751:
739:
710:
706:
696:
678:
670:
664:
660:
632:
628:
614:
605:
599:
593:
589:
583:
577:
573:
567:
558:
538:
527:
523:
516:
510:
502:
496:
488:
482:
474:
466:
458:
450:
443:
437:
433:
429:
426:
414:
408:
400:
394:
388:
382:
371:
363:
357:
349:
346:
329:
316:
309:
298:
295:Self-duality
286:such as the
284:affine space
281:
272:
262:
258:
256:
228:
208:
203:
197:
189:
183:
177:
173:
167:
159:
151:
143:
135:
127:
118:
113:
107:
103:
97:
91:
87:
80:
74:
67:
48:
42:
18:Desarguesian
944:, Blaisdell
744:Smith (1959
621:commutative
427:The points
379:the points
301:perspective
65:perspective
1314:Categories
1296:PlanetMath
919:, Oxford:
855:References
756:Katz (1998
55:, states:
1269:MathWorld
1252:EMS Press
1208:, Dover,
1068:120456855
1060:1432-1807
909:123036578
867:, Dover,
784:, pg. 19)
758:, p. 461)
746:, p. 307)
693:symmetric
421:collinear
354:If lines
75:centrally
61:triangles
1081:(1952),
952:(1969),
940:(1964),
814:), and (
719:See also
617:converse
320:converse
83:vertices
1010:0053531
1002:2031794
972:0123930
606:Pappian
564:hexagon
305:similar
253:History
68:axially
63:are in
1230:
1212:
1190:
1165:
1147:
1125:
1107:
1089:
1066:
1058:
1026:
1008:
1000:
970:
960:
927:
907:
871:
633:finite
629:finite
568:AbCaBc
282:In an
119:Axial
1064:S2CID
998:JSTOR
905:S2CID
731:Notes
235:field
209:This
1228:ISBN
1210:ISBN
1188:ISBN
1163:ISBN
1145:ISBN
1123:ISBN
1105:ISBN
1087:ISBN
1056:ISSN
1024:ISBN
958:ISBN
925:ISBN
869:ISBN
810:), (
675:and
615:The
597:and
581:and
507:and
471:and
455:and
441:and
419:are
405:and
368:and
194:and
164:and
148:and
132:and
111:and
95:and
59:Two
1303:at
1294:at
1285:at
1276:at
1267:at
1048:doi
990:doi
897:doi
697:any
237:or
176:.
117:.
43:In
1316::
1250:,
1244:,
1186:,
1077:;
1062:,
1054:,
1044:61
1042:,
1006:MR
1004:,
996:,
984:,
968:MR
966:,
923:,
903:,
893:47
891:,
885:,
818:).
711:A
695::
679:Cc
671:Bb
668:,
665:Aa
608:.
592:,
576:,
517:bc
514:∩
511:BC
503:ac
500:∩
497:AC
489:ab
486:∩
483:AB
475:ab
467:AB
459:Bb
451:Aa
436:,
432:,
415:bc
412:∩
409:BC
401:ac
398:∩
395:AC
392:,
389:ab
386:∩
383:AB
372:Cc
364:Bb
361:,
358:Aa
307:.
226:.
206:.
198:Cc
190:Bb
187:,
184:Aa
168:BC
160:bc
152:AC
144:ac
136:AB
128:ab
106:,
90:,
47:,
1050::
992::
986:4
899::
835:.
806:(
797:.
780:(
767:(
600:C
594:B
590:A
584:c
578:b
574:a
444:b
438:a
434:B
430:A
423:.
261:(
114:C
108:B
104:A
98:c
92:b
88:a
77:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
