Castillon

Castillon' s problem

Given a conic "c" and N (=4 here) points A, B, C, ... , inscribe a polygon in "c" whose sides A'B', B'C', ... pass respectively through the given points A, B, ... .

A simple solution [Berger: Geometry II, p. 181], can be obtained by considering the involutive homographies, preserving "c", whose Fregier points are the given points A, B, C, ... .
In the example above (N=4), there are four such homographies f1, f2, f3, f4, corresponding to A, B, C, and D. Their composition f = f4*f3*f2*f1 is again a well defined homography and has A' as a fixed point (f(A')=A'). Thus the location of A' can be determined by finding the fixed points of f (and intersecting their set with "c"). Once A' is known, the other points are determined by joining with A, finding then B' as intersection of "c" with AA' etc.

To see the construction of the involutive-homography, with given Fregier point look at : Fregier_Involutive.html .

To see the construction of the solution of the castillon's problem with c = circle and a triangle inscribed, look at : Castillon_Circle_Triangle.html

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