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orthocenter of a triangle is circumenter of another when triangles are similar

Source: Danube 2012 p2

July 22, 2019
geometrycircumcirclesimilar trianglesorthocenter

Problem Statement

Let ABCABC be an acute triangle and let A1A_1, B1B_1, C1C_1 be points on the sides BC,CABC, CA and ABAB, respectively. Show that the triangles ABCABC and A1B1C1A_1B_1C_1 are similar (∠A=∠A1,∠B=∠B1,∠C=∠C1\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1) if and only if the orthocentre of the triangle A1B1C1A_1B_1C_1 and the circumcentre of the triangle ABCABC coincide.
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