Question 158
Let $A X Y Z B$ be a convex pentagon inscribed in a semicircle with diameter $\overline{A B}$, and let $K$ be the foot of the altitude from $Y$ to $\overline{A B}$. Let $O$ denote the midpoint of $\overline{A B}$ and $L$ the intersection of $\overline{X Z}$ with $\overline{Y O}$. Select a point $M$ on line $K L$ with $M A = M B$, and finally, let $I$ be the reflection of $O$ across $\overline{X Z}$. Prove that if quadrilateral $X K O Z$ is cyclic then so is quadrilateral $Y O M I$.