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If exactly one pair of opposite sides of the hexagon are parallel, then the conclusion of the theorem is that the "Pascal line" determined by the two points of intersection is parallel to the parallel sides of the hexagon. If two pairs of opposite sides are parallel, then all three pairs of opposite
651:
that given any 8 points in general position, there is a unique ninth point such that all cubics through the first 8 also pass through the ninth point. In particular, if 2 general cubics intersect in 8 points then any other cubic through the same 8 points meets the ninth point of intersection of the
1224:
There exist 5-point, 4-point and 3-point degenerate cases of Pascal's theorem. In a degenerate case, two previously connected points of the figure will formally coincide and the connecting line becomes the tangent at the coalesced point. See the degenerate cases given in the added scheme and the
292:), which states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic; the conic may be degenerate, as in Pappus's theorem. The BraikenridgeâMaclaurin theorem may be applied in the
388:
It is sufficient to prove the theorem when the conic is a circle, because any (non-degenerate) conic can be reduced to a circle by a projective transformation. This was realised by Pascal, whose first lemma states the theorem for a circle. His second lemma states that what is true in one plane
1201:{\displaystyle {\frac {\overline {GB}}{\overline {GA}}}\times {\frac {\overline {HA}}{\overline {HF}}}\times {\frac {\overline {KF}}{\overline {KE}}}\times {\frac {\overline {GE}}{\overline {GD}}}\times {\frac {\overline {HD}}{\overline {HC}}}\times {\frac {\overline {KC}}{\overline {KB}}}=1.}
257:
are collinear in four points; the tangents being degenerate 'sides', taken at two possible positions on the 'hexagon' and the corresponding Pascal line sharing either degenerate intersection. This can be proven independently using a property of
156:
since any two lines meet and no exceptions need to be made for parallel lines. However, the theorem remains valid in the
Euclidean plane, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel.
60:, inscribed in a circle. Its sides are extended so that pairs of opposite sides intersect on Pascal's line. Each pair of extended opposite sides has its own color: one red, one yellow, one blue. Pascal's line is shown in white.
883:
Thus the group operation is associative. On the other hand, Pascal's theorem follows from the above associativity formula, and thus from the associativity of the group operation of elliptic curves by way of continuity.
340:
If six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This
356:
proved in 1849, these 60 lines can be associated with 60 points in such a way that each point is on three lines and each line contains three points. The 60 points formed in this way are now known as the
1544:
262:. If the conic is a circle, then another degenerate case says that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the
878:
389:
remains true upon projection to another plane. Degenerate conics follow by continuity (the theorem is true for non-degenerate conics, and thus holds in the limit of degenerate conic).
583:, where one of the points in the two tetrads overlap, hence meaning that other lines connecting the other three pairs must coincide to preserve cross ratio. Therefore,
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is also used to prove that the group operation on cubic elliptic curves is associative. The same group operation can be applied on a conic if we choose a point
1390:
1927:
293:
1742:
1465:
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a cubic and a conic have at most 3 × 2 = 6 points in common, unless they have a common component. So the cubic
610:. The proof makes use of the property that for every conic section we can find a one-sheet hyperboloid which passes through the conic.
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first two cubics. Pascal's theorem follows by taking the 8 points as the 6 points on the hexagon and two of the points (say,
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Again given the hexagon on a conic of Pascal's theorem with the above notation for points (in the first figure), we have
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in the figure). The first two cubics are two sets of 3 lines through the 6 points on the hexagon (for instance, the set
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1229:. If one chooses suitable lines of the Pascal-figures as lines at infinity one gets many interesting figures on
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which consist of a
Steiner point and three Kirkman points. The Steiner points also lie, four at a time, on 15
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133:, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.
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sides form pairs of parallel lines and there is no Pascal line in the
Euclidean plane (in this case, the
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sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in
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is the union of the conic and a line. It is now easy to check that this line is the Pascal line.
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will correspond to the isogonal conjugate if we overlap the similar triangles. This means that
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Six is the minimum number of points on a conic about which special statements can be made, as
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A short elementary computational proof in the case of the real projective plane was found by
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1625:, The Carus Mathematical Monographs, Number Four, The Mathematical Association of America
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Pascal's original note has no proof, but there are various modern proofs of the theorem.
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in a note written in 1639 when he was 16 years old and published the following year as a
27:
Theorem on the collinearity of three points generated from a hexagon inscribed on a conic
1601:
634:
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373:. Furthermore, the 20 Cayley lines pass four at a time through 15 points known as the
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Planar Circle
Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes
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in the figure) on the would-be Pascal line, and the ninth point as the third point (
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A short proof can be constructed using cross-ratio preservation. Projecting tetrad
103:
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A short elementary proof of Pascal's theorem in the case of a circle was found by
203:
A degenerate case of Pascal's theorem (four points) is interesting; given points
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of those points lie on a common line, the last point will be on that line, too.
400:). This proof proves the theorem for circle and then generalizes it to conics.
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1438:
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construction of the conic defined by five points, by varying the sixth point.
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There also exists a simple proof for Pascal's theorem for a circle using the
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has a component in common with the conic which must be the conic itself, so
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if necessary) meet at three points which lie on a straight line, called the
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17:
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Mills, Stella (March 1984), "Note on the
BraikenridgeâMaclaurin Theorem",
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1550:. NiedersÀchsiche Landesbibliothek, Gottfried Wilhelm Leibniz Bibliothek
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van Yzeren, Jan (1993), "A simple proof of Pascal's hexagon theorem",
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A very simple proof of Pascal's hexagon theorem and some applications
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629:
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of the extended
Euclidean plane is the Pascal line of the hexagon).
46:
inscribed in ellipse. Opposite sides of hexagon have the same color.
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233:, together with the intersection of tangents at opposite vertices
79:
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add up to the second intersection point of the conic with line
682:), and the third cubic is the union of the conic and the line
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is the cubic polynomial vanishing on the three lines through
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is obtained by first finding the intersection point of line
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cannot lie on the conic by genericity, and hence it lies on
1368:"A Property of Pascal's Hexagon Pascal May Have Overlooked"
633:
The intersections of the extended opposite sides of simple
1703:
by J. Chris Fisher and Norma Fuller (University of Regina)
1603:
The
Penguin Dictionary of Curious and Interesting Geometry
361:. The Pascal lines also pass, three at a time, through 20
1423:; Ryba, Alex (2012), "The Pascal Mysticum Demystified",
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is the second intersection point of the conic with line
152:
The most natural setting for Pascal's theorem is in a
86:) states that if six arbitrary points are chosen on a
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794:
592:
Another proof for Pascal's theorem for a circle uses
106:) and joined by line segments in any order to form a
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Interactive demo of Pascal's theorem (Java required)
1640:(10), Mathematical Association of America: 930â931,
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1803:
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1319:, p. 67 with a reference to Veblen and Young,
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1701:The Complete Pascal Figure Graphically Presented
1491:Notes and Records of the Royal Society of London
910:is the cubic vanishing on the other three lines
280:, named for 18th-century British mathematicians
1304:
1715:How to Project Spherical Conics into the Plane
1736:
647:Pascal's theorem has a short proof using the
602:, the geometer who discovered the celebrated
144:of two lines with three points on each line.
8:
1711:(PDF; 891 kB), Uni Darmstadt, S. 29â35.
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307:in 1847, as follows: suppose a polygon with
193:titled "Essay pour les coniques. Par B. P."
1391:Bulletin of the London Mathematical Society
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1525:Modenov, P.S.; Parkhomenko, A.S. (2001) ,
1477:, San Francisco, Calif.: HoldenâDay Inc.,
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643:(right) lie on the Pascal line MNP (left).
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196:Pascal's theorem is a special case of the
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873:{\displaystyle (A+B)+C=D+C=Q=A+F=A+(B+C)}
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410:We can infer the proof from existence of
1388:(1981), "T. P. Kirkman, mathematician",
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136:This theorem is a generalization of
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294:BraikenridgeâMaclaurin construction
110:, then the three pairs of opposite
1717:by Yoichi Maeda (Tokai University)
1473:Guggenheimer, Heinrich W. (1967),
122:of the hexagon. It is named after
25:
1633:The American Mathematical Monthly
1497:(2), The Royal Society: 235â240,
1212:Degenerations of Pascal's theorem
961:in common with the conic. But by
140:, which is the special case of a
129:The theorem is also valid in the
1891:
1882:
1881:
688:. Here the "ninth intersection"
1928:Theorems in projective geometry
1692:60 Pascal Lines (Java required)
1220:Pascal's theorem: degenerations
303:The theorem was generalized by
1426:The Mathematical Intelligencer
983:A property of Pascal's hexagon
867:
855:
807:
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278:BraikenridgeâMaclaurin theorem
1:
1585:Stefanovic, Nedeljko (2010),
1475:Plane geometry and its groups
955:is a cubic that has 7 points
271:five points determine a conic
1594:, Indian Academy of Sciences
1568:A Source Book in Mathematics
1565:Smith, David Eugene (1959),
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888:Proof using BĂ©zout's theorem
556:. This therefore means that
456:are collinear for concyclic
414:too. If we are to show that
75:hexagrammum mysticum theorem
1621:Young, John Wesley (1930),
1532:Encyclopedia of Mathematics
1954:
345:of 60 lines is called the
138:Pappus's (hexagon) theorem
1877:
1607:, London: Penguin Books,
1545:"Essay pour les coniques"
1439:10.1007/s00283-012-9301-4
1323:, vol. I, p. 138, Ex. 19.
719:in the plane. The sum of
396:, based on the proof in (
40:of self-crossing hexagon
1938:Euclidean plane geometry
922:on the conic and choose
713:on the conic and a line
705:CayleyâBacharach theorem
649:CayleyâBacharach theorem
625:Proof using cubic curves
532:, and projecting tetrad
198:CayleyâBacharach theorem
173:Pascal's theorem is the
1933:Theorems about polygons
1543:Pascal, Blaise (1640).
1211:
982:
916:. Pick a generic point
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305:August Ferdinand Möbius
185:. It was formulated by
1503:10.1098/rsnr.1984.0014
1333:Conway & Ryba 2012
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476:are similar, and that
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54:Self-crossing hexagon
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1599:Wells, David (1991),
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1812:Lettres provinciales
1404:10.1112/blms/13.2.97
1297:H. S. M. Coxeter
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792:
347:Hexagrammum Mysticum
335:Hexagrammum Mysticum
282:William Braikenridge
276:The converse is the
1775:Pascal's calculator
1623:Projective Geometry
1571:, New York: Dover,
1458:Greitzer, Samuel L.
1321:Projective Geometry
1252:Brianchon's theorem
1247:Desargues's theorem
958:A, B, C, D, E, F, P
550:, we obtain tetrad
526:, we obtain tetrad
462:, then notice that
183:Brianchon's theorem
72:(also known as the
66:projective geometry
1464:, Washington, DC:
1462:Geometry Revisited
1301:Samuel L. Greitzer
1257:Unicursal hexagram
1222:
1198:
928:so that the cubic
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608:Desargues' theorem
412:isogonal conjugate
148:Euclidean variants
102:in an appropriate
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1866:Marguerite PĂ©rier
1850:Jacqueline Pascal
1816:(1656–1657)
1790:Pascal's triangle
1454:Coxeter, H. S. M.
1227:circle geometries
1225:external link on
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405:Stefanovic (2010)
398:Guggenheimer 1967
394:van Yzeren (1993)
266:, are collinear.
264:Gergonne triangle
90:(which may be an
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142:degenerate conic
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1398:(2): 97â120,
1397:
1393:
1392:
1387:
1383:
1382:
1378:
1370:. 2014-02-03.
1369:
1363:
1360:
1357:, p. 172
1356:
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1299: and
1298:
1293:
1290:
1287:, p. 326
1286:
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408:
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401:
399:
395:
390:
386:
380:
378:
376:
375:Salmon points
372:
371:PlĂŒcker lines
368:
364:
360:
355:
350:
348:
344:
343:configuration
336:
333:
331:
328:
320:
312:
306:
301:
299:
296:, which is a
295:
291:
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254:
250:
242:
238:
231:
227:
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207:
201:
199:
194:
192:
188:
187:Blaise Pascal
184:
180:
176:
168:
166:
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147:
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139:
134:
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127:
125:
124:Blaise Pascal
121:
117:
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109:
105:
101:
97:
93:
89:
85:
82:for mystical
81:
77:
76:
71:
67:
58:
52:
44:
38:
32:
19:
1820:
1810:
1784:
1780:Pascal's law
1707:
1696:cut-the-knot
1687:cut-the-knot
1637:
1631:
1622:
1602:
1587:
1567:
1552:. Retrieved
1530:
1494:
1490:
1474:
1468:, p. 76
1461:
1430:
1424:
1421:Conway, John
1395:
1389:
1386:Biggs, N. L.
1362:
1350:
1339:
1328:
1320:
1312:
1292:
1226:
1223:
986:
974:
967:
957:
950:
944:
942:vanishes on
938:
934:
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918:
912:
906:
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894:
891:
882:
781:
775:
769:
763:
757:
751:
745:
739:
733:
727:
721:
715:
709:
702:
696:
690:
684:
678:
672:
666:
660:
654:
646:
639:
615:law of sines
612:
598:
596:repeatedly.
591:
585:
578:
574:
570:
566:
562:
558:
552:
546:
540:
534:
528:
522:
516:
510:
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484:
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465:
458:
452:
448:
444:
438:
434:
430:
424:
420:
416:
409:
402:
391:
387:
384:
374:
370:
367:Cayley lines
366:
362:
358:
351:
346:
339:
334:
326:
318:
310:
302:
275:
268:
252:
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236:
229:
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202:
195:
172:
159:
151:
135:
128:
119:
104:affine plane
74:
73:
69:
63:
56:
42:
36:
34:Pascal line
18:PlĂŒcker Line
1762:Innovations
1548:(facsimile)
1281:Pascal 1640
767:, which is
743:, which is
505:collinear.
209:on a conic
120:Pascal line
1912:Categories
1433:(3): 4â8,
1379:References
1355:Wells 1991
1344:Biggs 1981
1317:Young 1930
1285:Smith 1959
1235:hyperbolas
913:BC, DE, FA
901:AB, CD, EF
773:. Thus if
679:BC, DE, FA
673:AB, CD, EF
619:similarity
544:onto line
520:onto line
290:Mills 1984
260:pole-polar
1654:0002-9890
1537:EMS Press
1519:144663075
1447:122915551
1231:parabolas
1187:¯
1174:¯
1160:×
1154:¯
1141:¯
1127:×
1121:¯
1108:¯
1094:×
1088:¯
1075:¯
1061:×
1055:¯
1042:¯
1028:×
1022:¯
1009:¯
298:synthetic
191:broadside
100:hyperbola
1887:Category
1861:(sister)
1853:(sister)
1845:(father)
1460:(1967),
1241:See also
892:Suppose
637:hexagon
600:Dandelin
116:extended
96:parabola
84:hexagram
1897:Commons
1869:(niece)
1822:Pensées
1670:1252929
1662:2324214
1554:21 June
1483:0213943
1412:0608093
1303: (
948:. Then
785:, then
749:. Next
108:hexagon
92:ellipse
1835:Family
1826:(1669)
1765:Career
1668:
1660:
1652:
1611:
1575:
1517:
1511:531819
1509:
1481:
1445:
1410:
640:ABCDEF
635:cyclic
459:ABCDEF
381:Proofs
57:ABCDEF
43:ABCDEF
1804:Works
1658:JSTOR
1592:(PDF)
1515:S2CID
1507:JSTOR
1443:S2CID
1263:Notes
737:with
538:from
514:from
112:sides
88:conic
80:Latin
1650:ISSN
1609:ISBN
1573:ISBN
1556:2013
1305:1967
1233:and
904:and
755:and
725:and
703:The
658:and
617:and
569:) =
553:QBCY
535:ABCE
529:ABPX
511:ABCE
482:and
469:and
284:and
245:and
206:ABCD
177:and
1694:at
1685:at
1642:doi
1638:100
1499:doi
1435:doi
1400:doi
977:= 0
970:= 0
953:= 0
586:XYZ
502:XYZ
496:CYZ
494:= â
492:CYX
473:CYF
466:EYB
352:As
321:+ 1
313:+ 2
181:of
98:or
64:In
37:GHK
1914::
1666:MR
1664:,
1656:,
1648:,
1636:,
1535:,
1529:,
1513:,
1505:,
1495:38
1493:,
1479:MR
1456:;
1441:,
1431:34
1429:,
1408:MR
1406:,
1396:13
1394:,
1271:^
1237:.
1196:1.
939:λg
937:+
933:=
782:EN
764:EM
740:MP
734:AB
716:MP
700:.
697:MN
685:MN
621:.
579:CY
577:;
575:QB
567:PX
565:;
563:AB
547:BC
523:AB
453:FA
451:â©
449:CD
447:=
442:,
439:EF
437:â©
435:BC
433:=
428:,
425:DE
423:â©
421:AB
419:=
407:.
377:.
349:.
273:.
251:,
239:,
230:DA
228:â©
226:BC
223:,
220:CD
218:â©
216:AB
200:.
126:.
94:,
78:,
68:,
1744:e
1737:t
1730:v
1644::
1558:.
1501::
1437::
1402::
1307:)
1193:=
1183:B
1180:K
1170:C
1167:K
1150:C
1147:H
1137:D
1134:H
1117:D
1114:G
1104:E
1101:G
1084:E
1081:K
1071:F
1068:K
1051:F
1048:H
1038:A
1035:H
1018:A
1015:G
1005:B
1002:G
975:h
968:h
951:h
945:P
935:f
931:h
925:λ
919:P
907:g
895:f
868:)
865:C
862:+
859:B
856:(
853:+
850:A
847:=
844:F
841:+
838:A
835:=
832:Q
829:=
826:C
823:+
820:D
817:=
814:C
811:+
808:)
805:B
802:+
799:A
796:(
776:Q
770:D
758:B
752:A
746:M
728:B
722:A
710:E
691:P
667:P
661:N
655:M
581:)
573:(
571:R
561:(
559:R
541:F
517:D
490:â
485:Z
479:X
471:âł
464:âł
445:Z
431:Y
417:X
327:n
325:2
319:n
317:2
311:n
309:4
288:(
255:)
253:D
249:B
247:(
243:)
241:C
237:A
235:(
211:Î
20:)
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