Castillon_Circle

Castillon's problem

Given the circle "c" and three points A, B, C , construct triangle A'B'C', inscribed in "c" so that its sides pass through the given points: A'B' through A, B'C' through B and C'A' trhough C.

There are in general 2 solutions which are triangles A'B'C', A''B''C'' inversely oriented.

Given "c" and the points A,B,C we find the composition f=f3*f2*f1 of the Fregier Involutions with isolated fixed points (Fregier points) A, B, C respectively. This is done by taking three arbitrary points 1, 2, 3 on c and constructing their images 1', 2', 3' under f1, then the images of these under f2, which are 1'', 2'', 3'', then the images of these under f3, which are 1''', 2''', 3''' under f3. Finally we construct f by choosing the appropriate tool [Transforms \ Homogr. 1 conic _ ] and clicking on c and 1,1''', 2,2''' and 3,3''' (in that order). Having f (homography C*B*A), we right click on its tag and select [TransAss], holding simultaneously the Ctrl-key down. This defines the fixed points of f, two of them being F1 and F2. Here is the instability. In general there can be three distinct or one distinct and a whole line of fixed points. No three of them can be on the circle (otherwise f would be constant). Varying points A, B, C and/or the circle causes recalculation of the fixed points and F1, F2 can take the place of other fixed points, outside c.

http://users.math.uoc.gr/~pamfilos/