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Let ABCD be a convex quadrilateral whose vertices do not lie on a circle. Let A'B'C'D' be a quadrangle such that A',B', C',D' are the centers of the circumcircles of triangles BCD,ACD,ABD, and ABC. We write T (ABCD) = A'B'C'D'. Let us define A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).

(a) Prove that ABCD and A''B''C''D'' are similar.

(b) The ratio of similitude depends on the size of the angles of ABCD. Determine this ratio.

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