Baltic Way 2022 · Problem 12

An acute-angled triangle $ABC$ has altitudes $AD,BE$ and $CF$ (where $D,E$ and $F$ lie on the sides $BC,CA$ and $AB$, respectively). Let $Q$ be an interior point of the segment $AD$, and let the circumcircles of the triangles $QDF$ and $QDE$ meet the line $BC$ again at points $X$ and $Y$, respectively. Prove that $BX=CY$.