As we saw earlier, a conic section is the intersection of a (double-ended) right circular cone with a plane.
Here is a result proved by the French mathematician Michel Chasles (1793 to 1880) in 1852.
Let A, B, C, D be distinct points on a (non-singular) conic. If P is another point on the conic then the cross-ratio of the pencil PA, PB, PC, PD does not depend on the point P
By Exercises 8 Question 4, the cross-ratio of a pencil is the cross-ratio of the four points at which the pencil meets any line.
Project the conic into a circle. The cross ratio of the pencil is determined by the angles beween the lines PA, PB, PC, PD and if we move the point on the circle these angles are unchanged.
From this last result, if four points lie on a conic we may unambiguously talk about their cross-ratio.
We can now prove a result discovered by the French mathematician Blaise Pascal (1623 to 1662) when he was 16 years old. He called it the mysterium hexagrammicum = mystic hexagram.
If a hexagon is inscribed in a (non-singular) conic then the meets of opposite sides are collinear.
Project from 1: (2 , 4 ; 5 , 6) = (Q , 4 ; 5 , X)
Project from 3: (2 , 4 ; 5 , 6) = (P , Y ; 5 , 6)
Now consider the two pencils with vertex R and we get
(RQ , R4 ; R5 , RX) = (RP , RY ; R5 , R6)
But R6 = RX, R4 = RY and so RP = RQ and P, Q, R are collinear.
If a hexagon is circumscribed to a conic then the joins of opposite points of the hexagon are concurrent.