KöMaL 2017 A686

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 $X$.