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midpoint of CQ lies on \Gamma, right triangle and semicircle related

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August 4, 2019
geometrysemicirclemidpoint

Problem Statement

Let △ABC\vartriangle ABC be a right triangle with ∠B=90o\angle B = 90^o and ∣AB∣>∣BC∣|AB| > |BC|, and let Γ\Gamma be the semicircle with diameter ABAB that lies on the same side as CC. Let PP be a point on Γ\Gamma such that ∣BP∣=∣BC∣|BP| = |BC| and let QQ be on ABAB such that ∣AP∣=∣AQ∣|AP| = |AQ|. Prove that the midpoint of CQCQ lies on Γ\Gamma.
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