Question 43
Let $ABC$ be a triangle with $AB \lt AC$. The incircle of the triangle is tangent to $BC$ at $X$, to $CA$ at $Y$ and to $AB$ at $Z$. Let $U$ be the midpoint of the arc $\overarc{BC}$ containing $A$ of the circumcircle of triangle $ABC$. The line $UX$ meets the circumcircle again at $K$, and $AK$ meets $YZ$ at $T$. Prove that $XT$ is perpendicular to $YZ$.