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Similar triangles and parallelism

Source: OMEC Ecuador National Olympiad Final Round 2022 N3 P5 day 2

November 4, 2024
geometrysimilar trianglesnational olympiad

Problem Statement

Let ABCABC be a 90-degree triangle with hypotenuse BCBC. Let DD and EE distinct points on segment BCBC and P,QP, Q be the foot of the perpendicular from DD to ABAB and EE to ACAC, respectively. DPDP and EQEQ intersect at RR. Lines CRCR and ABAB intersect at MM and lines BRBR and ACAC intersect at NN. Prove that MN∥BCMN \parallel BC if and only if BD=CEBD=CE.
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