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Similarity through arc midpoint in right triangle

Source: Iranian Geometry Olympiad 2016 Medium 4

May 26, 2017
geometrycircumcircle

Problem Statement

Let ω\omega be the circumcircle of right-angled triangle ABCABC (∠A=90∘\angle A = 90^{\circ}). The tangent to ω\omega at point AA intersects the line BCBC at point PP. Suppose that MM is the midpoint of the minor arc ABAB, and PMPM intersects ω\omega for the second time in QQ. The tangent to ω\omega at point QQ intersects ACAC at KK. Prove that ∠PKC=90∘\angle PKC = 90^{\circ}.
Proposed by Davood Vakili
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