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IMO ShortList 1999, geometry problem 4

Source: IMO ShortList 1999, geometry problem 4

November 13, 2004
geometrytrigonometryTriangleanglessimilarityIMO Shortlist

Problem Statement

For a triangle T=ABCT = ABC we take the point XX on the side (AB)(AB) such that AX/AB=4/5AX/AB=4/5, the point YY on the segment (CX)(CX) such that CY=2YXCY = 2YX and, if possible, the point ZZ on the ray (CACA such that CXZ^=180−ABC^\widehat{CXZ} = 180 - \widehat{ABC}. We denote by Σ\Sigma the set of all triangles TT for which XYZ^=45\widehat{XYZ} = 45. Prove that all triangles from Σ\Sigma are similar and find the measure of their smallest angle.
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