The product H = G*F of the involutions is a homography with homography axis the line c = AB, which is the polar of intersection point C of lines a and b.

The proof is a simple case of Pascal's theorem. In fact, apply this to the hexagon of X

Note that in general a homography preserving a conic can be represented as a product of at most three involutions. This follows directly from the representation of a line homography as a product of at most three involutions. To see it conjugate if necessary the line homography F with an involution I

See the file InvolutionsProductGeneral.html for a discussion of the product of involutive homographies of the projective plane.

InvolutiveHomography.html

Pascal.html

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