3560:
4395:
4375:
4358:
4317:
3369:
3381:
3673:
1097:
3689:
20:
44:
3559:
3415:, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane. The proof of the closure of Thomsen's figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too:
4676:
A reason for using the notation above is that, for the ancient Greeks, a ratio is not a number or a geometrical object. We may think of ratio today as an equivalence class of pairs of geometrical objects. Also, equality for the Greeks is what we might today call congruence. In particular, distinct
4401:
The diagram for Lemma XIII is the same, but BA and DG, extended, meet at N. In any case, considering straight lines through G as cut by the three straight lines through A, (and accepting that equations of cross ratios remain valid after permutation of the entries,) we have by Lemma III or XI
4364:
Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear.
4310:
The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ).
4343:
of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular
3903:
4421:
Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear.
4368:
What we showed originally can be written as (J, ∞; K, L) = (J, G; D, B), with ∞ taking the place of the (nonexistent) intersection of JK and AG. Pappus shows this, in effect, in Lemma XI, whose diagram, however, has different lettering:
1434:
1819:
114:
4666:
Heath (Vol. II, p. 421) cites these propositions. The latter two can be understood as converses of the former two. Kline (p. 128) cites only
Proposition 139. The numbering of the propositions is as assigned by
2947:
The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician
4097:
Given three distinct points on each of two distinct lines, pair each point on one of the lines with one from the other line, then the joins of points not paired will meet in (opposite) pairs at points along a
870:
of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of
2157:
2826:
2616:
2296:
2956:. In general, Pappus's theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus's theorem does not hold are
4023:
4418:
Thus (E, H; J, G) = (E, K; D, L), so by Lemma X, the points H, M, and K are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear.
2437:
1517:
1313:
3516:
3618:
3212:
3155:
3266:
1975:
1929:
2009:
1104:
If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique.
1131:
804:
765:
1883:
1851:
1690:
3551:
1587:
1157:
3828:
3066:
4143:
1221:
2513:
2202:
1071:
966:
4087:
1470:
830:
726:
4259:
4201:
392:
203:
168:
4545:
2997:
2942:
2910:
2878:
2700:
2668:
2472:
2328:
1549:
1030:
998:
700:
668:
636:
326:
294:
235:
4509:
4483:
4049:
3742:
3716:
3644:
3317:
3096:
1339:
1189:
921:
895:
586:
4568:
4352:
It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X.
4282:
4224:
4166:
3360:
2723:
1656:
1633:
1610:
506:
483:
457:
434:
349:
262:
3802:
3782:
3762:
3664:
3409:
3337:
3289:
2846:
2636:
560:
526:
4602:
is not complete; he disregarded the possibility that some additional incidences could occur in the
Desargues configuration. A complete proof is provided by
4844:
Pambuccian, Victor; Schacht, Celia (2019), "The axiomatic destiny of the theorems of Pappus and
Desargues", in Dani, S. G.; Papadopoulos, A. (eds.),
4323:
Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also KL is drawn parallel to AZ. Then
1344:
1697:
4827:
3812:
If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear.
51:
3553:
One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first point.
4919:
2025:
2999:
happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference.
2728:
2518:
4853:
4713:
2207:
4300:. These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of
2880:. So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so
3912:
2336:
1475:
3808:
In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements:
4924:
4904:
4871:
1228:
3433:
4697:
4624:
3571:
3008:
924:
596:
3815:
Arranged in a matrix of nine points (as in the figure and description above) and thought of as evaluating a
3368:
3161:
3104:
3218:
3069:
856:
4394:
4374:
4357:
4316:
4051:
must be a line. Also, note that the same matrix formulation applies to the dual form of the theorem when
1934:
1888:
848:
1980:
3380:
1110:
770:
731:
3819:, if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear.
3898:{\displaystyle \left|{\begin{matrix}A&B&C\\a&b&c\\X&Y&Z\end{matrix}}\right|}
1856:
1824:
1666:
4895:
4293:
In its earliest known form, Pappus's
Theorem is Propositions 138, 139, 141, and 143 of Book VII of
3521:
1554:
867:
3293:
The left diagram shows the projective version, the right one an affine version, where the points
1136:
4327:
KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB).
3816:
3021:
533:
4111:
1159:. The key for a simple proof is the possibility for introducing a "suitable" coordinate system:
4512:
4339:
The last compound ratio (namely JD : GD & BG : JB) is what is known today as the
4102:
3672:
2961:
2953:
1194:
1085:
2477:
2166:
1035:
930:
4054:
1442:
809:
705:
4229:
4171:
362:
173:
138:
4723:
4705:
4518:
4410:
Considering straight lines through D as cut by the three straight lines through B, we have
4294:
3418:
Because the statement of
Thomsen's theorem (the closure of the figure) uses only the terms
3412:
2970:
2915:
2883:
2851:
2673:
2641:
2445:
2301:
1522:
1003:
971:
863:
673:
641:
609:
600:
359:. These three points are the points of intersection of the "opposite" sides of the hexagon
299:
267:
208:
124:
8:
4488:
4462:
4028:
3721:
3695:
3623:
3296:
3075:
1318:
1168:
900:
874:
844:
565:
459:
4550:
4264:
4206:
4148:
3342:
2705:
1638:
1615:
1592:
488:
465:
439:
416:
331:
244:
4801:
4747:
3787:
3767:
3747:
3649:
3394:
3322:
3274:
2949:
2831:
2621:
545:
511:
413:
If one considers a pappian plane containing a hexagon as just described but with sides
238:
4886:
4867:
4849:
4821:
4805:
4793:
4709:
4620:
4776:
Hessenberg, Gerhard (1905), "Beweis des
Desarguesschen Satzes aus dem Pascalschen",
4381:
What Pappus shows is DE.ZH : EZ.HD :: GB : BE, which we may write as
4785:
4739:
852:
604:
529:
399:
4719:
1082:
1429:{\displaystyle \;B=(0,\gamma ),\;C=(0,\delta ),\;\gamma ,\delta \notin \{0,1\}}
4913:
4797:
1814:{\displaystyle \;c=(0,0),\;b=(1,0),\;A=(0,1),\;B=(\gamma ,1),\;\gamma \neq 0}
1107:
Because of the parallelity in an affine plane one has to distinct two cases:
403:
4899:
4890:
2957:
1078:
840:
132:
4340:
1073:, the dual theorem is therefore just the same as the theorem itself. The
855:
into 2 straight lines. Pascal's theorem is in turn a special case of the
4105:
in at least two different ways, then they are perspective in three ways.
402:
over any field, but fails for projective planes over any noncommutative
4789:
4751:
3427:
1074:
109:{\displaystyle Ab\parallel aB,Bc\parallel bC\Rightarrow Ac\parallel aC}
3518:(see right diagram). The starting point of the sequence of chords is
352:
4743:
1096:
3688:
702:
defined by pairs of points resulting from pairs of intersections
19:
43:
4301:
406:. Projective planes in which the "theorem" is valid are called
2152:{\displaystyle C=(1,0,0),\;c=(0,1,0),\;X=(0,0,1),\;A=(1,1,1)}
3391:
If in the affine version of the dual "little theorem" point
3362:
than one gets the "dual little theorem" of Pappus' theorem.
2960:
projective planes over noncommutative division rings, and
2821:{\displaystyle x_{2}=x_{1}q,\;x_{1}=x_{3}p,\;x_{3}=x_{2}r}
2611:{\displaystyle x_{1}=x_{2}p,\;x_{2}=x_{3}q,\;x_{3}=x_{1}r}
4730:
Cronheim, A. (1953), "A proof of
Hessenberg's theorem",
3430:
invariant, and one can introduce coordinates such that
4331:
These proportions might be written today as equations:
2291:{\displaystyle x_{2}=x_{3},\;x_{1}=x_{3},\;x_{2}=x_{1}}
3837:
1489:
4553:
4521:
4491:
4465:
4267:
4232:
4209:
4174:
4151:
4114:
4057:
4031:
3915:
3831:
3790:
3770:
3750:
3724:
3698:
3652:
3626:
3574:
3524:
3436:
3397:
3345:
3325:
3299:
3277:
3221:
3164:
3107:
3078:
3024:
2973:
2918:
2886:
2854:
2834:
2731:
2708:
2676:
2644:
2624:
2521:
2480:
2448:
2339:
2304:
2210:
2169:
2028:
2014:
1983:
1937:
1891:
1859:
1827:
1700:
1669:
1641:
1618:
1595:
1557:
1525:
1478:
1445:
1347:
1321:
1231:
1197:
1171:
1139:
1113:
1038:
1006:
974:
933:
903:
877:
812:
773:
734:
708:
676:
644:
612:
568:
548:
514:
491:
468:
442:
419:
365:
334:
302:
270:
247:
211:
176:
141:
54:
4598:, pg. 159, footnote 1), Hessenberg's original proof
1225:In this case coordinates are introduced, such that
4018:{\displaystyle \ ABC,abc,AbZ,BcX,CaY,XbC,YcA,ZaB\ }
3646:) as dual theorem of the little theorem of Pappus (
4905:Pappus’s Theorem: Nine proofs and three variations
4562:
4539:
4503:
4477:
4276:
4253:
4218:
4195:
4160:
4137:
4081:
4043:
4017:
3897:
3796:
3776:
3756:
3736:
3710:
3683:
3658:
3638:
3612:
3545:
3510:
3403:
3354:
3331:
3311:
3283:
3260:
3206:
3149:
3090:
3060:
2991:
2936:
2904:
2872:
2840:
2820:
2717:
2694:
2662:
2630:
2610:
2507:
2466:
2431:
2322:
2290:
2196:
2151:
2003:
1969:
1923:
1877:
1845:
1813:
1694:In this case the coordinates are chosen such that
1684:
1650:
1627:
1604:
1581:
1543:
1511:
1464:
1428:
1333:
1307:
1215:
1183:
1151:
1125:
1065:
1024:
992:
960:
915:
889:
824:
798:
759:
720:
694:
662:
630:
580:
554:
520:
500:
477:
451:
428:
386:
343:
320:
288:
256:
229:
197:
162:
108:
4843:
4837:Mathematical Thought From Ancient to Modern Times
2432:{\displaystyle B=(p,1,1),\;Y=(1,q,1),\;b=(1,1,r)}
1512:{\displaystyle b=({\tfrac {\delta }{\gamma }},0)}
35:are collinear on the Pappus line. The hexagon is
4911:
4732:Proceedings of the American Mathematical Society
1308:{\displaystyle \;S=(0,0),\;A=(0,1),\;c=(1,0)\;}
588:have a point in common, one gets the so-called
3271:are concurrent, that means: they have a point
836:means that the lines pass through one point.)
4814:Pappi Alexandrini Collectionis Quae Supersunt
4784:(2), Berlin / Heidelberg: Springer: 161–172,
4025:are lines, then Pappus's theorem states that
1977:, respectively, and at least the parallelity
4677:line segments may be equal. Ratios are not
1423:
1411:
4644:
4642:
3511:{\displaystyle P=(0,0),\;Q=(1,0),\;R=(0,1)}
4826:: CS1 maint: location missing publisher (
4775:
4599:
4134:
4115:
3486:
3461:
3009:principle of duality for projective planes
2791:
2761:
2581:
2551:
2401:
2370:
2264:
2237:
2121:
2090:
2059:
2000:
1984:
1966:
1938:
1920:
1892:
1801:
1776:
1751:
1726:
1701:
1455:
1398:
1373:
1348:
1304:
1282:
1257:
1232:
786:
747:
4757:
4595:
4578:
4576:
1000:of the dual theorem, and collinearity of
4729:
4639:
4603:
4335:KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB).
3687:
1095:
42:
18:
4861:
4811:
4696:
4582:
3613:{\displaystyle \color {red}1,2,3,4,5,6}
4912:
4573:
3207:{\displaystyle Y:=(c\cap A)(C\cap a),}
3150:{\displaystyle X:=(A\cap b)(a\cap B),}
2618:, so they pass through the same point
638:, and another set of concurrent lines
4834:
4766:
3411:is a point at infinity too, one gets
3261:{\displaystyle Z:=(b\cap C)(B\cap c)}
2952:proved that Pappus's theorem implies
2019:Choose homogeneous coordinates with
1091:
1088:graph with 18 vertices and 27 edges.
2670:. The condition for the three lines
170:and another set of collinear points
4839:, New York: Oxford University Press
4681:in this sense; but they may be the
1970:{\displaystyle \;a=(\gamma +1,0)\;}
1924:{\displaystyle \;C=(\gamma +1,1)\;}
1077:of the Pappus configuration is the
13:
2015:Proof with homogeneous coordinates
2004:{\displaystyle \;Ac\parallel Ca\;}
1439:From the parallelity of the lines
927:. Since, in particular, the lines
508:parallel (so that the Pappus line
14:
4936:
4880:
3575:
3319:are points at infinity. If point
1519:and the parallelity of the lines
1126:{\displaystyle g\not \parallel h}
968:have the properties of the lines
799:{\displaystyle a\cap C,\;B\cap c}
760:{\displaystyle a\cap B,\;A\cap c}
23:Pappus's hexagon theorem: Points
4896:Dual to Pappus's hexagon theorem
4393:
4373:
4356:
4315:
4092:are triples of concurrent lines.
3671:
3558:
3379:
3367:
3068:are chosen alternately from two
1032:is equivalent to concurrence of
4920:Theorems in projective geometry
4698:Coxeter, Harold Scott MacDonald
4670:
4660:
3684:Other statements of the theorem
3002:
2828:to pass through the same point
4848:, Springer, pp. 355–399,
4771:, New York: Dover Publications
4769:A History of Greek Mathematics
4651:
4630:
4609:
4588:
4459:However, this does occur when
4453:
4440:
4431:
4389:The diagram for Lemma XII is:
4076:
4058:
3537:
3525:
3505:
3493:
3480:
3468:
3455:
3443:
3255:
3243:
3240:
3228:
3198:
3186:
3183:
3171:
3141:
3129:
3126:
3114:
2426:
2408:
2395:
2377:
2364:
2346:
2146:
2128:
2115:
2097:
2084:
2066:
2053:
2035:
1963:
1945:
1917:
1899:
1878:{\displaystyle cB\parallel bC}
1846:{\displaystyle Ab\parallel Ba}
1795:
1783:
1770:
1758:
1745:
1733:
1720:
1708:
1685:{\displaystyle g\parallel h\ }
1576:
1564:
1506:
1485:
1392:
1380:
1367:
1355:
1301:
1289:
1276:
1264:
1251:
1239:
88:
1:
4690:
4450:, BI-Taschenbuch, 1969, p. 93
3546:{\displaystyle (0,\lambda ).}
3374:dual theorem: projective form
1582:{\displaystyle a=(\delta ,0)}
603:states that given one set of
592:version of Pappus's theorem.
539:shown in the second diagram.
205:then the intersection points
47:Pappus's theorem: affine form
4812:Hultsch, Fridericus (1877),
4385:(D, Z; E, H) = (∞, B; E, G).
4348:(J, G; D, B) = (J, Z; H, E).
3666:is at infinity, too !).
1152:{\displaystyle g\parallel h}
7:
4414:(L, D; E, K) = (G, D; ∞ Z).
4406:(G, J; E, H) = (G, D; ∞ Z).
3061:{\displaystyle A,b,C,a,B,c}
10:
4941:
4704:(2nd ed.), New York:
4288:
4138:{\displaystyle \;AB,CD,\;}
1821:. From the parallelity of
4866:, Rudolph Steiner Press,
4762:, Berlin: Springer-Verlag
4758:Dembowski, Peter (1968),
4619:, Springer-Verlag, 2013,
3386:dual theorem: affine form
1216:{\displaystyle S=g\cap h}
923:. This configuration is
4925:Euclidean plane geometry
4887:Pappus's hexagon theorem
4702:Introduction to Geometry
4448:Grundlagen der Geometrie
4425:
2967:The proof is invalid if
2508:{\displaystyle XB,CY,cb}
2197:{\displaystyle AC,Ac,AX}
1066:{\displaystyle Bc,bC,XY}
961:{\displaystyle Bc,bC,XY}
857:Cayley–Bacharach theorem
121:Pappus's hexagon theorem
4862:Whicher, Olive (1971),
4767:Heath, Thomas (1981) ,
4082:{\displaystyle (A,B,C)}
2962:non-Desarguesian planes
1465:{\displaystyle Bc,\;Cb}
825:{\displaystyle b\cap C}
721:{\displaystyle A\cap b}
4835:Kline, Morris (1972),
4564:
4541:
4505:
4479:
4278:
4255:
4254:{\displaystyle AD,BE,}
4220:
4197:
4196:{\displaystyle DE,FA,}
4162:
4139:
4083:
4045:
4019:
3899:
3805:
3798:
3778:
3758:
3738:
3712:
3660:
3640:
3614:
3547:
3512:
3405:
3356:
3333:
3313:
3285:
3262:
3208:
3151:
3092:
3062:
3013:dual theorem of Pappus
2993:
2938:
2906:
2874:
2842:
2822:
2719:
2696:
2664:
2632:
2612:
2509:
2468:
2433:
2324:
2292:
2198:
2153:
2005:
1971:
1925:
1879:
1847:
1815:
1686:
1652:
1629:
1606:
1583:
1545:
1513:
1466:
1430:
1335:
1309:
1217:
1185:
1153:
1127:
1101:
1067:
1026:
994:
962:
917:
891:
839:Pappus's theorem is a
826:
800:
761:
722:
696:
664:
632:
582:
556:
522:
502:
479:
453:
430:
388:
387:{\displaystyle AbCaBc}
345:
322:
290:
258:
231:
199:
198:{\displaystyle a,b,c,}
164:
163:{\displaystyle A,B,C,}
116:
110:
40:
4778:Mathematische Annalen
4706:John Wiley & Sons
4565:
4542:
4540:{\displaystyle Aa,Bb}
4506:
4480:
4279:
4256:
4226:are concurrent, then
4221:
4198:
4163:
4140:
4101:If two triangles are
4084:
4046:
4020:
3900:
3799:
3779:
3759:
3744:are perspective from
3739:
3713:
3691:
3678:Thomsen figure: proof
3661:
3641:
3615:
3548:
3513:
3406:
3357:
3334:
3314:
3286:
3263:
3209:
3152:
3093:
3063:
2994:
2992:{\displaystyle C,c,X}
2939:
2937:{\displaystyle X,Y,Z}
2907:
2905:{\displaystyle pq=qp}
2875:
2873:{\displaystyle rpq=1}
2843:
2823:
2720:
2697:
2695:{\displaystyle Cb,cB}
2665:
2663:{\displaystyle rqp=1}
2633:
2613:
2510:
2469:
2467:{\displaystyle p,q,r}
2434:
2325:
2323:{\displaystyle B,Y,b}
2293:
2199:
2154:
2006:
1972:
1926:
1880:
1848:
1816:
1687:
1653:
1635:and is parallel line
1630:
1607:
1584:
1546:
1544:{\displaystyle Ab,Ba}
1514:
1467:
1431:
1341:have the coordinates
1336:
1310:
1218:
1186:
1154:
1128:
1100:Pappus theorem: proof
1099:
1068:
1027:
1025:{\displaystyle X,Y,Z}
995:
993:{\displaystyle x,y,z}
963:
918:
892:
827:
801:
762:
723:
697:
695:{\displaystyle x,y,z}
665:
663:{\displaystyle a,b,c}
633:
631:{\displaystyle A,B,C}
583:
557:
523:
503:
480:
454:
431:
389:
346:
323:
321:{\displaystyle aC,Bc}
291:
289:{\displaystyle aB,Ac}
259:
232:
230:{\displaystyle X,Y,Z}
200:
165:
111:
46:
22:
4617:Projektive Geometrie
4551:
4519:
4489:
4463:
4265:
4230:
4207:
4172:
4149:
4112:
4055:
4029:
3913:
3829:
3788:
3784:, and so, also from
3768:
3748:
3722:
3696:
3650:
3624:
3572:
3522:
3434:
3395:
3343:
3323:
3297:
3275:
3219:
3162:
3105:
3076:
3022:
2971:
2916:
2884:
2852:
2832:
2729:
2706:
2674:
2642:
2622:
2519:
2478:
2446:
2337:
2302:
2208:
2167:
2026:
1981:
1935:
1889:
1857:
1825:
1698:
1667:
1639:
1616:
1593:
1555:
1523:
1476:
1443:
1345:
1319:
1229:
1195:
1169:
1137:
1111:
1036:
1004:
972:
931:
901:
875:
864:Pappus configuration
810:
771:
732:
706:
674:
642:
610:
566:
546:
512:
489:
466:
440:
417:
363:
332:
300:
268:
245:
209:
174:
139:
125:Pappus of Alexandria
52:
4864:Projective Geometry
4846:Geometry in history
4657:Whicher, chapter 14
4504:{\displaystyle abc}
4478:{\displaystyle ABC}
4168:are concurrent and
4044:{\displaystyle XYZ}
3737:{\displaystyle BbY}
3711:{\displaystyle XcC}
3639:{\displaystyle PQR}
3426:, the statement is
3312:{\displaystyle G,H}
3091:{\displaystyle G,H}
2954:Desargues's theorem
1334:{\displaystyle B,C}
1191:intersect at point
1184:{\displaystyle g,h}
916:{\displaystyle abc}
890:{\displaystyle ABC}
581:{\displaystyle g,h}
542:If the Pappus line
537:of Pappus's theorem
4790:10.1007/BF01457558
4563:{\displaystyle Cc}
4560:
4537:
4501:
4475:
4437:Coxeter, pp. 236–7
4277:{\displaystyle CF}
4274:
4251:
4219:{\displaystyle BC}
4216:
4193:
4161:{\displaystyle EF}
4158:
4135:
4079:
4041:
4015:
3895:
3889:
3806:
3794:
3774:
3754:
3734:
3708:
3656:
3636:
3610:
3609:
3543:
3508:
3420:connect, intersect
3401:
3355:{\displaystyle GH}
3352:
3329:
3309:
3281:
3258:
3204:
3147:
3088:
3058:
2989:
2950:Gerhard Hessenberg
2934:
2902:
2870:
2838:
2818:
2718:{\displaystyle XY}
2715:
2692:
2660:
2628:
2608:
2505:
2474:. The three lines
2464:
2429:
2320:
2298:, take the points
2288:
2194:
2149:
2001:
1967:
1921:
1875:
1843:
1811:
1682:
1651:{\displaystyle Ac}
1648:
1628:{\displaystyle -1}
1625:
1605:{\displaystyle Ca}
1602:
1579:
1541:
1509:
1498:
1462:
1426:
1331:
1305:
1213:
1181:
1149:
1123:
1102:
1092:Proof: affine form
1063:
1022:
990:
958:
913:
887:
822:
796:
757:
718:
692:
660:
628:
578:
552:
518:
501:{\displaystyle bC}
498:
478:{\displaystyle Bc}
475:
452:{\displaystyle aB}
449:
429:{\displaystyle Ab}
426:
384:
344:{\displaystyle bC}
341:
318:
286:
257:{\displaystyle Ab}
254:
227:
195:
160:
117:
106:
41:
4855:978-3-030-13611-6
4760:Finite Geometries
4715:978-0-471-50458-0
4600:Hessenberg (1905)
4446:Rolf Lingenberg:
4014:
3918:
3797:{\displaystyle Z}
3777:{\displaystyle a}
3757:{\displaystyle A}
3659:{\displaystyle U}
3413:Thomsen's theorem
3404:{\displaystyle U}
3332:{\displaystyle U}
3284:{\displaystyle U}
2841:{\displaystyle Z}
2631:{\displaystyle a}
1692:(little theorem).
1681:
1497:
853:conic degenerates
832:are concurrent. (
670:, then the lines
601:incidence theorem
555:{\displaystyle u}
521:{\displaystyle u}
131:given one set of
4932:
4876:
4858:
4840:
4831:
4825:
4817:
4808:
4772:
4763:
4754:
4726:
4685:
4674:
4668:
4664:
4658:
4655:
4649:
4646:
4637:
4634:
4628:
4613:
4607:
4592:
4586:
4580:
4571:
4569:
4567:
4566:
4561:
4546:
4544:
4543:
4538:
4510:
4508:
4507:
4502:
4484:
4482:
4481:
4476:
4457:
4451:
4444:
4438:
4435:
4397:
4377:
4360:
4319:
4283:
4281:
4280:
4275:
4260:
4258:
4257:
4252:
4225:
4223:
4222:
4217:
4202:
4200:
4199:
4194:
4167:
4165:
4164:
4159:
4144:
4142:
4141:
4136:
4088:
4086:
4085:
4080:
4050:
4048:
4047:
4042:
4024:
4022:
4021:
4016:
4012:
3916:
3904:
3902:
3901:
3896:
3894:
3890:
3803:
3801:
3800:
3795:
3783:
3781:
3780:
3775:
3763:
3761:
3760:
3755:
3743:
3741:
3740:
3735:
3717:
3715:
3714:
3709:
3675:
3665:
3663:
3662:
3657:
3645:
3643:
3642:
3637:
3620:of the triangle
3619:
3617:
3616:
3611:
3562:
3552:
3550:
3549:
3544:
3517:
3515:
3514:
3509:
3410:
3408:
3407:
3402:
3383:
3371:
3361:
3359:
3358:
3353:
3338:
3336:
3335:
3330:
3318:
3316:
3315:
3310:
3290:
3288:
3287:
3282:
3267:
3265:
3264:
3259:
3213:
3211:
3210:
3205:
3156:
3154:
3153:
3148:
3097:
3095:
3094:
3089:
3067:
3065:
3064:
3059:
2998:
2996:
2995:
2990:
2943:
2941:
2940:
2935:
2912:. Equivalently,
2911:
2909:
2908:
2903:
2879:
2877:
2876:
2871:
2847:
2845:
2844:
2839:
2827:
2825:
2824:
2819:
2814:
2813:
2801:
2800:
2784:
2783:
2771:
2770:
2754:
2753:
2741:
2740:
2724:
2722:
2721:
2716:
2701:
2699:
2698:
2693:
2669:
2667:
2666:
2661:
2637:
2635:
2634:
2629:
2617:
2615:
2614:
2609:
2604:
2603:
2591:
2590:
2574:
2573:
2561:
2560:
2544:
2543:
2531:
2530:
2514:
2512:
2511:
2506:
2473:
2471:
2470:
2465:
2438:
2436:
2435:
2430:
2329:
2327:
2326:
2321:
2297:
2295:
2294:
2289:
2287:
2286:
2274:
2273:
2260:
2259:
2247:
2246:
2233:
2232:
2220:
2219:
2203:
2201:
2200:
2195:
2158:
2156:
2155:
2150:
2010:
2008:
2007:
2002:
1976:
1974:
1973:
1968:
1930:
1928:
1927:
1922:
1884:
1882:
1881:
1876:
1852:
1850:
1849:
1844:
1820:
1818:
1817:
1812:
1691:
1689:
1688:
1683:
1679:
1657:
1655:
1654:
1649:
1634:
1632:
1631:
1626:
1611:
1609:
1608:
1603:
1588:
1586:
1585:
1580:
1550:
1548:
1547:
1542:
1518:
1516:
1515:
1510:
1499:
1490:
1471:
1469:
1468:
1463:
1435:
1433:
1432:
1427:
1340:
1338:
1337:
1332:
1315:(see diagram).
1314:
1312:
1311:
1306:
1222:
1220:
1219:
1214:
1190:
1188:
1187:
1182:
1158:
1156:
1155:
1150:
1132:
1130:
1129:
1124:
1086:distance-regular
1072:
1070:
1069:
1064:
1031:
1029:
1028:
1023:
999:
997:
996:
991:
967:
965:
964:
959:
922:
920:
919:
914:
896:
894:
893:
888:
847:for a conic—the
845:Pascal's theorem
831:
829:
828:
823:
805:
803:
802:
797:
766:
764:
763:
758:
727:
725:
724:
719:
701:
699:
698:
693:
669:
667:
666:
661:
637:
635:
634:
629:
605:concurrent lines
587:
585:
584:
579:
561:
559:
558:
553:
532:), one gets the
530:line at infinity
527:
525:
524:
519:
507:
505:
504:
499:
484:
482:
481:
476:
458:
456:
455:
450:
435:
433:
432:
427:
400:projective plane
393:
391:
390:
385:
350:
348:
347:
342:
327:
325:
324:
319:
295:
293:
292:
287:
263:
261:
260:
255:
236:
234:
233:
228:
204:
202:
201:
196:
169:
167:
166:
161:
119:In mathematics,
115:
113:
112:
107:
16:Geometry theorem
4940:
4939:
4935:
4934:
4933:
4931:
4930:
4929:
4910:
4909:
4883:
4874:
4856:
4819:
4818:
4744:10.2307/2031794
4716:
4693:
4688:
4675:
4671:
4665:
4661:
4656:
4652:
4648:Coxeter, p. 233
4647:
4640:
4636:Coxeter, p. 231
4635:
4631:
4614:
4610:
4593:
4589:
4581:
4574:
4570:are concurrent.
4552:
4549:
4548:
4520:
4517:
4516:
4490:
4487:
4486:
4464:
4461:
4460:
4458:
4454:
4445:
4441:
4436:
4432:
4428:
4291:
4284:are concurrent.
4266:
4263:
4262:
4231:
4228:
4227:
4208:
4205:
4204:
4173:
4170:
4169:
4150:
4147:
4146:
4113:
4110:
4109:
4056:
4053:
4052:
4030:
4027:
4026:
3914:
3911:
3910:
3888:
3887:
3882:
3877:
3871:
3870:
3865:
3860:
3854:
3853:
3848:
3843:
3836:
3832:
3830:
3827:
3826:
3789:
3786:
3785:
3769:
3766:
3765:
3749:
3746:
3745:
3723:
3720:
3719:
3697:
3694:
3693:
3686:
3679:
3676:
3667:
3651:
3648:
3647:
3625:
3622:
3621:
3573:
3570:
3569:
3563:
3523:
3520:
3519:
3435:
3432:
3431:
3396:
3393:
3392:
3387:
3384:
3375:
3372:
3344:
3341:
3340:
3339:is on the line
3324:
3321:
3320:
3298:
3295:
3294:
3292:
3276:
3273:
3272:
3220:
3217:
3216:
3163:
3160:
3159:
3106:
3103:
3102:
3077:
3074:
3073:
3023:
3020:
3019:
3007:Because of the
3005:
2972:
2969:
2968:
2944:are collinear.
2917:
2914:
2913:
2885:
2882:
2881:
2853:
2850:
2849:
2833:
2830:
2829:
2809:
2805:
2796:
2792:
2779:
2775:
2766:
2762:
2749:
2745:
2736:
2732:
2730:
2727:
2726:
2725:with equations
2707:
2704:
2703:
2675:
2672:
2671:
2643:
2640:
2639:
2638:if and only if
2623:
2620:
2619:
2599:
2595:
2586:
2582:
2569:
2565:
2556:
2552:
2539:
2535:
2526:
2522:
2520:
2517:
2516:
2479:
2476:
2475:
2447:
2444:
2443:
2338:
2335:
2334:
2303:
2300:
2299:
2282:
2278:
2269:
2265:
2255:
2251:
2242:
2238:
2228:
2224:
2215:
2211:
2209:
2206:
2205:
2168:
2165:
2164:
2027:
2024:
2023:
2017:
1982:
1979:
1978:
1936:
1933:
1932:
1890:
1887:
1886:
1858:
1855:
1854:
1826:
1823:
1822:
1699:
1696:
1695:
1693:
1668:
1665:
1664:
1640:
1637:
1636:
1617:
1614:
1613:
1594:
1591:
1590:
1556:
1553:
1552:
1524:
1521:
1520:
1488:
1477:
1474:
1473:
1444:
1441:
1440:
1346:
1343:
1342:
1320:
1317:
1316:
1230:
1227:
1226:
1224:
1196:
1193:
1192:
1170:
1167:
1166:
1138:
1135:
1134:
1112:
1109:
1108:
1094:
1037:
1034:
1033:
1005:
1002:
1001:
973:
970:
969:
932:
929:
928:
902:
899:
898:
876:
873:
872:
811:
808:
807:
772:
769:
768:
733:
730:
729:
707:
704:
703:
675:
672:
671:
643:
640:
639:
611:
608:
607:
567:
564:
563:
547:
544:
543:
513:
510:
509:
490:
487:
486:
467:
464:
463:
462:and also sides
441:
438:
437:
418:
415:
414:
364:
361:
360:
355:, lying on the
333:
330:
329:
301:
298:
297:
269:
266:
265:
246:
243:
242:
210:
207:
206:
175:
172:
171:
140:
137:
136:
127:) states that
123:(attributed to
53:
50:
49:
48:
17:
12:
11:
5:
4938:
4928:
4927:
4922:
4908:
4907:
4902:
4893:
4882:
4881:External links
4879:
4878:
4877:
4872:
4859:
4854:
4841:
4832:
4809:
4773:
4764:
4755:
4738:(2): 219–221,
4727:
4714:
4692:
4689:
4687:
4686:
4669:
4659:
4650:
4638:
4629:
4608:
4596:Dembowski 1968
4594:According to (
4587:
4572:
4559:
4556:
4536:
4533:
4530:
4527:
4524:
4500:
4497:
4494:
4474:
4471:
4468:
4452:
4439:
4429:
4427:
4424:
4416:
4415:
4408:
4407:
4399:
4398:
4387:
4386:
4379:
4378:
4362:
4361:
4350:
4349:
4337:
4336:
4329:
4328:
4321:
4320:
4290:
4287:
4286:
4285:
4273:
4270:
4250:
4247:
4244:
4241:
4238:
4235:
4215:
4212:
4192:
4189:
4186:
4183:
4180:
4177:
4157:
4154:
4133:
4130:
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4124:
4121:
4118:
4106:
4099:
4094:
4093:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4040:
4037:
4034:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3907:
3906:
3905:
3893:
3886:
3883:
3881:
3878:
3876:
3873:
3872:
3869:
3866:
3864:
3861:
3859:
3856:
3855:
3852:
3849:
3847:
3844:
3842:
3839:
3838:
3835:
3821:
3820:
3813:
3793:
3773:
3753:
3733:
3730:
3727:
3707:
3704:
3701:
3685:
3682:
3681:
3680:
3677:
3670:
3668:
3655:
3635:
3632:
3629:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3566:Thomsen figure
3564:
3557:
3542:
3539:
3536:
3533:
3530:
3527:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3400:
3389:
3388:
3385:
3378:
3376:
3373:
3366:
3351:
3348:
3328:
3308:
3305:
3302:
3280:
3269:
3268:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3214:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3157:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3087:
3084:
3081:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3004:
3001:
2988:
2985:
2982:
2979:
2976:
2933:
2930:
2927:
2924:
2921:
2901:
2898:
2895:
2892:
2889:
2869:
2866:
2863:
2860:
2857:
2837:
2817:
2812:
2808:
2804:
2799:
2795:
2790:
2787:
2782:
2778:
2774:
2769:
2765:
2760:
2757:
2752:
2748:
2744:
2739:
2735:
2714:
2711:
2691:
2688:
2685:
2682:
2679:
2659:
2656:
2653:
2650:
2647:
2627:
2607:
2602:
2598:
2594:
2589:
2585:
2580:
2577:
2572:
2568:
2564:
2559:
2555:
2550:
2547:
2542:
2538:
2534:
2529:
2525:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2463:
2460:
2457:
2454:
2451:
2440:
2439:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2319:
2316:
2313:
2310:
2307:
2285:
2281:
2277:
2272:
2268:
2263:
2258:
2254:
2250:
2245:
2241:
2236:
2231:
2227:
2223:
2218:
2214:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2161:
2160:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2016:
2013:
1999:
1996:
1993:
1990:
1987:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1874:
1871:
1868:
1865:
1862:
1842:
1839:
1836:
1833:
1830:
1810:
1807:
1804:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1678:
1675:
1672:
1647:
1644:
1624:
1621:
1601:
1598:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1540:
1537:
1534:
1531:
1528:
1508:
1505:
1502:
1496:
1493:
1487:
1484:
1481:
1461:
1458:
1454:
1451:
1448:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1330:
1327:
1324:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1212:
1209:
1206:
1203:
1200:
1180:
1177:
1174:
1148:
1145:
1142:
1122:
1119:
1116:
1093:
1090:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1021:
1018:
1015:
1012:
1009:
989:
986:
983:
980:
977:
957:
954:
951:
948:
945:
942:
939:
936:
912:
909:
906:
886:
883:
880:
821:
818:
815:
795:
792:
789:
785:
782:
779:
776:
756:
753:
750:
746:
743:
740:
737:
717:
714:
711:
691:
688:
685:
682:
679:
659:
656:
653:
650:
647:
627:
624:
621:
618:
615:
577:
574:
571:
562:and the lines
551:
535:affine version
517:
497:
494:
474:
471:
448:
445:
425:
422:
408:pappian planes
398:It holds in a
396:
395:
383:
380:
377:
374:
371:
368:
340:
337:
317:
314:
311:
308:
305:
285:
282:
279:
276:
273:
253:
250:
226:
223:
220:
217:
214:
194:
191:
188:
185:
182:
179:
159:
156:
153:
150:
147:
144:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
75:
72:
69:
66:
63:
60:
57:
15:
9:
6:
4:
3:
2:
4937:
4926:
4923:
4921:
4918:
4917:
4915:
4906:
4903:
4901:
4897:
4894:
4892:
4888:
4885:
4884:
4875:
4873:0-85440-245-4
4869:
4865:
4860:
4857:
4851:
4847:
4842:
4838:
4833:
4829:
4823:
4815:
4810:
4807:
4803:
4799:
4795:
4791:
4787:
4783:
4779:
4774:
4770:
4765:
4761:
4756:
4753:
4749:
4745:
4741:
4737:
4733:
4728:
4725:
4721:
4717:
4711:
4707:
4703:
4699:
4695:
4694:
4684:
4680:
4673:
4663:
4654:
4645:
4643:
4633:
4626:
4622:
4618:
4615:W. Blaschke:
4612:
4605:
4604:Cronheim 1953
4601:
4597:
4591:
4584:
4579:
4577:
4557:
4554:
4534:
4531:
4528:
4525:
4522:
4514:
4498:
4495:
4492:
4472:
4469:
4466:
4456:
4449:
4443:
4434:
4430:
4423:
4419:
4413:
4412:
4411:
4405:
4404:
4403:
4396:
4392:
4391:
4390:
4384:
4383:
4382:
4376:
4372:
4371:
4370:
4366:
4359:
4355:
4354:
4353:
4347:
4346:
4345:
4342:
4334:
4333:
4332:
4326:
4325:
4324:
4318:
4314:
4313:
4312:
4308:
4307:
4303:
4299:
4296:
4271:
4268:
4248:
4245:
4242:
4239:
4236:
4233:
4213:
4210:
4190:
4187:
4184:
4181:
4178:
4175:
4155:
4152:
4131:
4128:
4125:
4122:
4119:
4116:
4107:
4104:
4100:
4096:
4095:
4091:
4073:
4070:
4067:
4064:
4061:
4038:
4035:
4032:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3908:
3891:
3884:
3879:
3874:
3867:
3862:
3857:
3850:
3845:
3840:
3833:
3825:
3824:
3823:
3822:
3818:
3814:
3811:
3810:
3809:
3791:
3771:
3751:
3731:
3728:
3725:
3705:
3702:
3699:
3690:
3674:
3669:
3653:
3633:
3630:
3627:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3567:
3561:
3556:
3555:
3554:
3540:
3534:
3531:
3528:
3502:
3499:
3496:
3490:
3487:
3483:
3477:
3474:
3471:
3465:
3462:
3458:
3452:
3449:
3446:
3440:
3437:
3429:
3425:
3421:
3416:
3414:
3398:
3382:
3377:
3370:
3365:
3364:
3363:
3349:
3346:
3326:
3306:
3303:
3300:
3278:
3252:
3249:
3246:
3237:
3234:
3231:
3225:
3222:
3215:
3201:
3195:
3192:
3189:
3180:
3177:
3174:
3168:
3165:
3158:
3144:
3138:
3135:
3132:
3123:
3120:
3117:
3111:
3108:
3101:
3100:
3099:
3098:, the lines
3085:
3082:
3079:
3072:with centers
3071:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3016:
3014:
3010:
3000:
2986:
2983:
2980:
2977:
2974:
2965:
2963:
2959:
2955:
2951:
2945:
2931:
2928:
2925:
2922:
2919:
2899:
2896:
2893:
2890:
2887:
2867:
2864:
2861:
2858:
2855:
2835:
2815:
2810:
2806:
2802:
2797:
2793:
2788:
2785:
2780:
2776:
2772:
2767:
2763:
2758:
2755:
2750:
2746:
2742:
2737:
2733:
2712:
2709:
2689:
2686:
2683:
2680:
2677:
2657:
2654:
2651:
2648:
2645:
2625:
2605:
2600:
2596:
2592:
2587:
2583:
2578:
2575:
2570:
2566:
2562:
2557:
2553:
2548:
2545:
2540:
2536:
2532:
2527:
2523:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2461:
2458:
2455:
2452:
2449:
2423:
2420:
2417:
2414:
2411:
2405:
2402:
2398:
2392:
2389:
2386:
2383:
2380:
2374:
2371:
2367:
2361:
2358:
2355:
2352:
2349:
2343:
2340:
2333:
2332:
2331:
2317:
2314:
2311:
2308:
2305:
2283:
2279:
2275:
2270:
2266:
2261:
2256:
2252:
2248:
2243:
2239:
2234:
2229:
2225:
2221:
2216:
2212:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2163:On the lines
2143:
2140:
2137:
2134:
2131:
2125:
2122:
2118:
2112:
2109:
2106:
2103:
2100:
2094:
2091:
2087:
2081:
2078:
2075:
2072:
2069:
2063:
2060:
2056:
2050:
2047:
2044:
2041:
2038:
2032:
2029:
2022:
2021:
2020:
2012:
1997:
1994:
1991:
1988:
1985:
1960:
1957:
1954:
1951:
1948:
1942:
1939:
1914:
1911:
1908:
1905:
1902:
1896:
1893:
1872:
1869:
1866:
1863:
1860:
1840:
1837:
1834:
1831:
1828:
1808:
1805:
1802:
1798:
1792:
1789:
1786:
1780:
1777:
1773:
1767:
1764:
1761:
1755:
1752:
1748:
1742:
1739:
1736:
1730:
1727:
1723:
1717:
1714:
1711:
1705:
1702:
1676:
1673:
1670:
1663:
1659:
1645:
1642:
1622:
1619:
1599:
1596:
1589:. Hence line
1573:
1570:
1567:
1561:
1558:
1538:
1535:
1532:
1529:
1526:
1503:
1500:
1494:
1491:
1482:
1479:
1459:
1456:
1452:
1449:
1446:
1437:
1420:
1417:
1414:
1408:
1405:
1402:
1399:
1395:
1389:
1386:
1383:
1377:
1374:
1370:
1364:
1361:
1358:
1352:
1349:
1328:
1325:
1322:
1298:
1295:
1292:
1286:
1283:
1279:
1273:
1270:
1267:
1261:
1258:
1254:
1248:
1245:
1242:
1236:
1233:
1210:
1207:
1204:
1201:
1198:
1178:
1175:
1172:
1164:
1160:
1146:
1143:
1140:
1120:
1117:
1114:
1105:
1098:
1089:
1087:
1084:
1080:
1076:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1019:
1016:
1013:
1010:
1007:
987:
984:
981:
978:
975:
955:
952:
949:
946:
943:
940:
937:
934:
926:
910:
907:
904:
884:
881:
878:
869:
868:configuration
865:
860:
858:
854:
850:
849:limiting case
846:
842:
837:
835:
819:
816:
813:
793:
790:
787:
783:
780:
777:
774:
754:
751:
748:
744:
741:
738:
735:
715:
712:
709:
689:
686:
683:
680:
677:
657:
654:
651:
648:
645:
625:
622:
619:
616:
613:
606:
602:
598:
593:
591:
575:
572:
569:
549:
540:
538:
536:
531:
515:
495:
492:
472:
469:
461:
446:
443:
423:
420:
411:
409:
405:
404:division ring
401:
381:
378:
375:
372:
369:
366:
358:
354:
338:
335:
315:
312:
309:
306:
303:
283:
280:
277:
274:
271:
251:
248:
240:
224:
221:
218:
215:
212:
192:
189:
186:
183:
180:
177:
157:
154:
151:
148:
145:
142:
134:
130:
129:
128:
126:
122:
103:
100:
97:
94:
91:
85:
82:
79:
76:
73:
70:
67:
64:
61:
58:
55:
45:
38:
34:
30:
26:
21:
4900:cut-the-knot
4891:cut-the-knot
4863:
4845:
4836:
4813:
4781:
4777:
4768:
4759:
4735:
4731:
4701:
4682:
4678:
4672:
4662:
4653:
4632:
4616:
4611:
4590:
4583:Coxeter 1969
4455:
4447:
4442:
4433:
4420:
4417:
4409:
4400:
4388:
4380:
4367:
4363:
4351:
4338:
4330:
4322:
4309:
4305:
4297:
4292:
4089:
3909:That is, if
3807:
3565:
3423:
3419:
3417:
3390:
3270:
3017:
3012:
3006:
3003:Dual theorem
2966:
2958:Desarguesian
2946:
2441:
2162:
2018:
1661:
1660:
1438:
1162:
1161:
1106:
1103:
1079:Pappus graph
861:
841:special case
838:
833:
594:
589:
541:
534:
412:
407:
397:
356:
120:
118:
36:
32:
28:
24:
4515:, that is,
4513:perspective
4341:cross ratio
4103:perspective
3018:If 6 lines
2204:, given by
357:Pappus line
4914:Categories
4691:References
4625:3034869320
4298:Collection
3692:Triangles
3291:in common.
1612:has slope
1165:The lines
1075:Levi graph
834:Concurrent
4806:120456855
4798:1432-1807
3817:permanent
3535:λ
3250:∩
3235:∩
3193:∩
3178:∩
3136:∩
3121:∩
3015:is true:
2442:for some
1992:∥
1949:γ
1903:γ
1885:one gets
1867:∥
1835:∥
1806:≠
1803:γ
1787:γ
1674:∥
1620:−
1568:δ
1495:γ
1492:δ
1472:one gets
1409:∉
1406:δ
1400:γ
1390:δ
1365:γ
1208:∩
1144:∥
1083:bipartite
925:self dual
851:when the
817:∩
791:∩
778:∩
752:∩
739:∩
713:∩
353:collinear
133:collinear
98:∥
89:⇒
80:∥
62:∥
4822:citation
4816:, Berlin
4700:(1969),
4667:Hultsch.
4627:, S. 190
4585:, p. 238
4306:Porisms.
4295:Pappus's
3568:(points
3428:affinely
3424:parallel
1118:∦
599:of this
460:parallel
4752:2031794
4724:0123930
4511:are in
4289:Origins
3070:pencils
2330:to be
1662:Case 2:
1551:yields
1163:Case 1:
866:is the
528:is the
135:points
4870:
4852:
4804:
4796:
4750:
4722:
4712:
4623:
4302:Euclid
4013:
3917:
1680:
590:little
241:pairs
37:AbCaBc
4802:S2CID
4748:JSTOR
4683:same.
4679:equal
4426:Notes
4098:line.
4868:ISBN
4850:ISBN
4828:link
4794:ISSN
4710:ISBN
4621:ISBN
4547:and
4485:and
4261:and
4203:and
4145:and
4090:etc.
3764:and
3718:and
3422:and
3011:the
2702:and
2515:are
1931:and
1853:and
1133:and
1081:, a
897:and
862:The
806:and
767:and
728:and
597:dual
595:The
485:and
436:and
351:are
328:and
296:and
264:and
239:line
31:and
4898:at
4889:at
4786:doi
4740:doi
4304:'s
4108:If
2848:is
843:of
237:of
4916::
4824:}}
4820:{{
4800:,
4792:,
4782:61
4780:,
4746:,
4734:,
4720:MR
4718:,
4708:,
4641:^
4575:^
3226::=
3169::=
3112::=
2964:.
2011:.
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4059:(
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3892:|
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3834:|
3804:.
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3532:,
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3466:=
3463:Q
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3456:)
3453:0
3450:,
3447:0
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3441:=
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3350:H
3347:G
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3301:G
3279:U
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3253:c
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3241:)
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3223:Z
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2773:=
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2713:Y
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2687:c
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2658:1
2655:=
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2500:c
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2393:1
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2375:=
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2344:=
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2315:,
2312:Y
2309:,
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2284:1
2280:x
2276:=
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2262:,
2257:3
2253:x
2249:=
2244:1
2240:x
2235:,
2230:3
2226:x
2222:=
2217:2
2213:x
2192:X
2189:A
2186:,
2183:c
2180:A
2177:,
2174:C
2171:A
2159:.
2147:)
2144:1
2141:,
2138:1
2135:,
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2119:,
2116:)
2113:1
2110:,
2107:0
2104:,
2101:0
2098:(
2095:=
2092:X
2088:,
2085:)
2082:0
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2076:1
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2070:0
2067:(
2064:=
2061:c
2057:,
2054:)
2051:0
2048:,
2045:0
2042:,
2039:1
2036:(
2033:=
2030:C
1998:a
1995:C
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1961:0
1958:,
1955:1
1952:+
1946:(
1943:=
1940:a
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1915:1
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1909:1
1906:+
1900:(
1897:=
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1873:C
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1832:b
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1809:0
1799:,
1796:)
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1774:,
1771:)
1768:1
1765:,
1762:0
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1756:=
1753:A
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1746:)
1743:0
1740:,
1737:1
1734:(
1731:=
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1724:,
1721:)
1718:0
1715:,
1712:0
1709:(
1706:=
1703:c
1677:h
1671:g
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1643:A
1623:1
1600:a
1597:C
1577:)
1574:0
1571:,
1565:(
1562:=
1559:a
1539:a
1536:B
1533:,
1530:b
1527:A
1507:)
1504:0
1501:,
1486:(
1483:=
1480:b
1460:b
1457:C
1453:,
1450:c
1447:B
1424:}
1421:1
1418:,
1415:0
1412:{
1403:,
1396:,
1393:)
1387:,
1384:0
1381:(
1378:=
1375:C
1371:,
1368:)
1362:,
1359:0
1356:(
1353:=
1350:B
1329:C
1326:,
1323:B
1302:)
1299:0
1296:,
1293:1
1290:(
1287:=
1284:c
1280:,
1277:)
1274:1
1271:,
1268:0
1265:(
1262:=
1259:A
1255:,
1252:)
1249:0
1246:,
1243:0
1240:(
1237:=
1234:S
1223:.
1211:h
1205:g
1202:=
1199:S
1179:h
1176:,
1173:g
1147:h
1141:g
1121:h
1115:g
1061:Y
1058:X
1055:,
1052:C
1049:b
1046:,
1043:c
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1020:Z
1017:,
1014:Y
1011:,
1008:X
988:z
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979:,
976:x
956:Y
953:X
950:,
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944:b
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938:c
935:B
911:c
908:b
905:a
885:C
882:B
879:A
820:C
814:b
794:c
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784:,
781:C
775:a
755:c
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745:,
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690:z
687:,
684:y
681:,
678:x
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655:,
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623:,
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394:.
382:c
379:B
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370:b
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339:C
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313:B
310:,
307:C
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284:c
281:A
278:,
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225:Z
222:,
219:Y
216:,
213:X
193:,
190:c
187:,
184:b
181:,
178:a
158:,
155:C
152:,
149:B
146:,
143:A
104:C
101:a
95:c
92:A
86:C
83:b
77:c
74:B
71:,
68:B
65:a
59:b
56:A
39:.
33:Z
29:Y
25:X
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