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UV is parallel to BC, weird mixtilinear condition

Source: 2023 Israel Olympic Revenge P2

June 15, 2023
olympic revengegeometrymixtilinear

Problem Statement

Triangle ΔABC\Delta ABC is inscribed in circle Ω\Omega. The tangency point of Ω\Omega and the AA-mixtilinear circle of ΔABC\Delta ABC is TT. Points EE, FF were chosen on ACAC, ABAB respectively so that EF∥BCEF\parallel BC and (TEF)(TEF) is tangent to Ω\Omega. Let ω\omega denote the AA-excircle of ΔAEF\Delta AEF, which is tangent to sides EFEF, AEAE, AFAF at KK, YY, ZZ respectively. Line ATAT intersects ω\omega at two points PP, QQ with PP between AA and QQ. Let QKQK and YZYZ intersect at VV, and let the tangent to ω\omega at PP and the tangent to Ω\Omega at TT intersect at UU. Prove that UV∥BCUV\parallel BC.
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