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Very nice geo problem of ISL 1986

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August 31, 2010
geometrycircumcircleratiotrigonometryIMO Shortlist

Problem Statement

Let ABCDABCD be a convex quadrilateral whose vertices do not lie on a circle. Let A′B′C′D′A'B'C'D' be a quadrangle such that A′,B′,C′,D′A',B', C',D' are the centers of the circumcircles of triangles BCD,ACD,ABDBCD,ACD,ABD, and ABCABC. We write T(ABCD)=A′B′C′D′T (ABCD) = A'B'C'D'. Let us define A′′B′′C′′D′′=T(A′B′C′D′)=T(T(ABCD)).A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).
(a) Prove that ABCDABCD and A′′B′′C′′D′′A''B''C''D'' are similar.
(b) The ratio of similitude depends on the size of the angles of ABCDABCD. Determine this ratio.
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