Baltic Way 2022 · Problem 15

Let $\Omega$ be a circle, and $B\ne C$ two fixed points on $\Omega$. Given a third point $A$ on $\Omega$, let $X$ and $Y$ denote the feet of the altitudes from $B$ and $C$, respectively, in the triangle $ABC$. Prove that there exists a circle $\Gamma$ such that $XY$ is tangent to $\Gamma$ regardless of the choice of the point $A$.