Question 25
Let $ABC$ be an acute triangle, and let $AA_{1}$, $BB_{1}$, and $CC_{1}$ be its altitudes. Segments $AA_{1}$ and $B_{1}C_{1}$ meet at point $K$. The perpendicular bisector of segment $A_{1}K$ intersects sides $AB$ and $AC$ at $L$ and $M$, respectively. Prove that points $A$, $A_{1}$, $L$, and $M$ lie on a circle.