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Show that the measure of the angle AZX does not depend on the choice of X.

Source: 2010 Peru Iberoamerican TST problem 5

May 10, 2023
geometry

Problem Statement

The trapeze ABCDABCD with bases ABAB and CDCD is inscribed in a circle Γ\Gamma. Let XX be a variable point of the arc \overarc{AB} that does not contain either CC or DD. Let YY be the point of intersection of ABAB and DXDX, and let ZZ be the point of the segment CXCX such that XZXC=AYAB\frac{XZ}{XC}=\frac{AY}{AB}. Prove that the measure of the angle ∠AZX\angle AZX does not depend on the choice of XX.
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