Question 7
Let $ABC$ be a triangle inscribed in circle $(O)$ such that two points $B$, $C$ are fixed, $A$ moves on major arc $BC$ of $(O)$. Two tangents through $B$, $C$ of $(O)$ intersect at $P$. Circle with diameter $OP$ intersects $AC$, $AB$ at $D$, $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.