Involutions product

The context is that of homographies (in particular involutive) preserving a conic. Here we study the product of two involutions F(A, a) and G(B, b). In parenthesis are the center (Fregier point) of the involution and its homography axis. The property is:
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 X1, X2, Y1, Y2, Z1, Z2. The intersection points are A(X1Y1, X2Y2), D(X1Z2,X2Z1), B(Y1Z1,Y2Z2). D is on line AB. Analogously E is on line AB, thus AB coincides with the homography axis of the composition G*F. Note that F*G is the inverse homography having the same homography axis.
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 Ix, so that the product F'=Ix*F fixes the point at infinity. Then this is a translation or a similarity which decomposes trivialy in a product of involutions. Everything transfers to conic homographies via good parametrizations.
See the file InvolutionsProductGeneral.html for a discussion of the product of involutive homographies of the projective plane.