Pascal's Theorem may be restated as

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.

Then the converse is plainly

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 and E.

The figure below was drawn with Cabri and displayed using CabriJava.

If you drag the red line XY, F moves, but always lies on the yellow curve.
This is the conic determined by A,B,C,D,E.
Here, you may also move A,B,C,D or E to vary the conic.

Notes
(1) The condition on A,B,C,D,E is to ensure that the conic is non-degenerate.
(2) The proof is quite short, though tricky. You can find it here.
(3) Pascal's Theorem and its Converse give rise to an interesting construction.

Pascal's Theorem

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