This is one of the gems of Projective Geometry.

Pascal's Theorem
Suppose that the points A,B,C,D,E,F lie on a non-degenerate conic.
Suppose also that AC, DB meet in X, CF, BE in Y and FD, EA in Z.
Then X, Y and Z are collinear. The points A,B,C,D,E are fixed, so determine a conic, here an ellipse.
The point F is on this conic, but may be moved to any position.
To vary F, move the cursor to the point F. Holding down the left button
F may be dragged round the ellipse.
The red line is the line XY. As you move F, observe that Z is always on XY.
This is precisely what the theorem asserts.

The figure was drawn with Cabri and displayed using CabriJava.
See the Technical Notes below for more details.

Mathematical Notes
(1) We can think of X,Y,Z as the intersections of opposite sides of the hexagon ACFDBE.
(2) The order of the points round the conic is immaterial. In the figure we initially have
the points in the cyclic order ABCDEF.
(3) The theorem holds for any non-degenerate conic. The figure shows an ellipse.
(4) If the conic degenerates into a pair of lines, and A,B,F lie on one line and C,D,E on the
other, the result still holds. This is Pappus's Theorem.
(5) A proof, with necessary prerequisites, can be found here.
(6) The converse of Pascal's Theorem is valid, and has a useful consequence.

Historical Note
Blaise Pascal discovered this theorem in 1640 - at the age of 16.

Technical Notes
Cabri is a dynamic geometry program. It allows the construction of geometrical figures.
It is dynamic since the geometrical objects may be moved. When the point F above is moved,
the point Y moves so that it is the intersection of CF and BE. The line XY is redrawn. The point
Z moves so it is the intersection of FD and EA. Thus, everything that depends on F is updated.
CabriJava is a piece of Java code which displays Cabri figures on the Web. The viewer may
interact with the figure, as in Cabri, but cannot create new objects.