BALTIC-WAY 2022 Problem 12

An acute-angled triangle $ABC$ has altitudes $AD, BE$ and $CF$ . 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 $BX = CY$ .