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XY _|_ AM if RX, SY _|_ DE, MD = ME, AB, AC tangents of (O)

Source: 2021 Saudi Arabia Training Lists p6 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 4, 2022
geometryperpendicularTangents

Problem Statement

Let AA be a point lies outside circle (O)(O) and tangent lines ABAB, ACAC of (O)(O). Consider points D,E,MD, E, M on (O)(O) such that MD=MEMD = ME. The line DEDE cuts MBMB, MCMC at R,SR, S. Take X∈OBX \in OB, Y∈OCY \in OC such that RX,SY⊥DERX, SY \perp DE. Prove that XY⊥AMXY \perp AM.
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