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Circle with lots of symmetry

Source: Latvia BW TST 2021 P10

September 26, 2021
geometrysymmetry

Problem Statement

Circle ω\omega with centre MM and diameter XYXY is given. Point AA is chosen on ω\omega so that AX<AYAX<AY. Points B,CB, C are chosen on segments XM,YMXM, YM, respectively, in a way that BM=CMBM=CM. A parallel line to ABAB is constructed through CC; the line intersects ω\omega at PP so that PP lies on the smaller arc AY^\widehat{AY}. Similarly, a parallel line to ACAC is constructed through BB; the line intersects ω\omega at QQ so that QQ lies on the smaller arc XA^\widehat{XA}. Lines PQPQ and XYXY intersect at SS. Prove that ASAS is tangent to ω\omega.
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