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perpendicular wanted, arc midpoint, tangent lines to circles

Source: 2018 Saudi Arabia GMO TST II p3

August 1, 2020
geometryperpendicularTangentsarc midpoint

Problem Statement

Let CC be a point lies outside the circle (O)(O) and CS,CTCS, CT are tangent lines of (O)(O). Take two points A,BA, B on (O)(O) with MM is the midpoint of the minor arc ABAB such that A,B,MA, B, M differ from S,TS, T. Suppose that MS,MTMS, MT cut line ABAB at E,FE, F. Take X∈OSX \in OS and Y∈OTY \in OT such that EX,FYEX, FY are perpendicular to ABAB. Prove that XYX Y and CMC M are perpendicular.
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