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3 equal circumcircles

Source: Mexican Mathematical Olympiad 2005 OMM P1

July 31, 2018
geometrycircumcirclecircumradius

Problem Statement

Let OO be the center of the circumcircle of an acute triangle ABCABC, let PP be any point inside the segment BCBC. Suppose the circumcircle of triangle BPOBPO intersects the segment ABAB at point RR and the circumcircle of triangle COPCOP intersects CACA at point QQ. (i) Consider the triangle PQRPQR, show that it is similar to triangle ABCABC and that OO is its orthocenter. (ii) Show that the circumcircles of triangles BPOBPO, COPCOP, PQRPQR have the same radius.
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