Pascal's Theorem and its Converse lead to an interesting construction for points on the conic determined by five points with no three collinear.
This appears to have been discovered independently by Braikenridge and Maclaurin. There was a dispute between the two over priority. There
is an account of the dispute on the St Andrews history pages. What is clear is that Braikenridge published first (in 1733), and knew of the
construction by 1726. Maclaurin quoted it in lectures around 1725, and may have known it several years earlier.
Pascal's Theorem Suppose that no three of the points A,B,C,D,E, are collinear. If the point F lies on the conic determined by A,B,C,D and E, then X = ACnDB, Y = CFnBE, Z = FDnEA are collinear. |
The Converse Suppose that no three of the points A,B,C,D,E, are collinear. If X = ACnDB, Y = CFnBE, Z = FDnEA are collinear, then the point F lies on the conic determined by A,B,C,D,E. |
The Construction Suppose that no three of the points A,B,C,D,E, are collinear. Let L be any line through X = ACnDB. Suppose that L meets BE in Y, and EA in Z. Let F be the interection of CY and DZ. Then (1) F lies on the conic C determined by A,B,C,D,E, and (2) any point on C is the point F for some line L through X. |
Together, (1) and (2) show that the conic is the locus of the points produced by the family of lines through X = ACnDB. |
Verification of the Construction To show (1), we use the Converse of Pascal's Theorem. When F is so defined, Y is the intersection of CF=CY and BE, and Z the intersection of DF=DZ and EA. As X,Y,Z are on L, they are collinear, and the Converse gives F on C. To show (2), we use Pascal's Theorem. If F is on C, X = ACnDB, Y = CFnBE, Z = FDnEA are collinear. Thus F is produced by the construction using the line L = XY. |
Pascal's Theorem |
Converse of Pascal | Main Geometry Page |