Question 102
Let $\omega$ be the incircle of a triangle $ABC$ (with $AB \lt AC$). Let the excircle $\omega_A$ touch the side $BC$ at $A'$. A variable point $X$ is chosen on the segment $AA'$ so that the segment $A'X$ has no common points with $\omega$. The tangents to $\omega$ passing through $X$ meet the side $BC$ at $Y$ and $Z$. Prove that the sum $XY + XZ$ does not depend on the choice of $X$. (I. Mitrofanov)