KöMaL problem A, 2017

Geometry Difficulty 7.0 National Olympiad, round 2 Prove it Hungary

In the triangle $ABC$ , the incircle is denoted by $\omega$ . The excircle opposite to $A$ is tangent to the line $BC$ at point $A_1$ . Let $X$ be an arbitrarily chosen point on the extension of the segment $AA_1$ beyond $A_1$ , and let the line $BC$ meet the tangents drawn from $X$ to $\omega$ at $Y$ and $Z$ , such that $BY<BZ$ . Show that $XY-XZ$ does not depend on the position of the point $X$ .

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