Pascal’s Theorem #

Pascal’s theorem (Blaise Pascal, 1640 — the hexagrammum mysticum) states:

If six points lie on a conic, then the three pairs of opposite sides of the hexagon they form meet in three collinear points. The line containing them is the Pascal line.

It is the projective generalization of Pappus’s theorem (which is the special case where the conic degenerates into two lines), and it is dual to Brianchon’s theorem.

As a special case of the 8-conic theorem #

In the Penrose configuration, Pascal’s theorem arises by letting the cube’s conics degenerate so that they collapse onto a single conic together with the sides of an inscribed hexagon. The shared-point condition on the chords of contact in each face becomes exactly the collinearity asserted by Pascal: the three intersection points of opposite hexagon sides line up on the Pascal line.

In the applet below, the white curve is the conic; the colored lines are the six hexagon sides; and the dotted line is the Pascal line on which the three opposite-side intersections lie.

You can drag the configuration directly in the view. The collinearity of the three opposite-side intersections — the content of Pascal’s theorem — is preserved throughout.