PAMO 2025 Problem 1

Let $\triangle{ABC}$ be a triangle. A circle passing through $B$ and $C$ intersects $AB$ and $AC$ at points $D$ and $E$ respectively. The orthogonal projections of points $B$ and $E$ onto $CD$ are noted respectively $B'$ and $E'$ . The orthogonal projections of points $D$ and $C$ onto $BE$ are noted respectively $D'$ and $C'$ . Show that points $B'$ , $C'$ , $D'$ and $E'$ lie on the same circle.