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Another geo P1

Source: Balkan MO 2022 P1

May 6, 2022
geometrycircumcircleBalkan Mathematics Olympiadprojective geometry

Problem Statement

Let ABCABC be an acute triangle such that CA≠CBCA \neq CB with circumcircle ω\omega and circumcentre OO. Let tAt_A and tBt_B be the tangents to ω\omega at AA and BB respectively, which meet at XX. Let YY be the foot of the perpendicular from OO onto the line segment CXCX. The line through CC parallel to line ABAB meets tAt_A at ZZ. Prove that the line YZYZ passes through the midpoint of the line segment ACAC.
Proposed by Dominic Yeo, United Kingdom
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