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The sum not depends to x

Source: All russian olympiad 2016,Day2,grade 9,P7

May 1, 2016
geometrycircumcircleincircle

Problem Statement

In triangle ABCABC,AB<ACAB<AC and ω\omega is incirle.The AA-excircle is tangent to BCBC at A′A^\prime.Point XX lies on AA′AA^\prime such that segment A′XA^\prime X doesn't intersect with ω\omega.The tangents from XX to ω\omega intersect with BCBC at Y,ZY,Z.Prove that the sum XY+XZXY+XZ not depends to point XX.(Mitrofanov)
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