Question 70
Fix the point $C$ on a circle centred at $O$ with diameter $A B$, with $C$ distinct from both $A$ and $B$. Now let $D$ vary among all points of the arc $A B$ not containing $C$ that are distinct from both $A$ and $B$. Given $D$, let $E$ be the point on the segment $C D$ such that the lines $B E$ and $C D$ are perpendicular.
Prove that the product $C E \cdot E D$ attains its maximum, as $D$ varies, precisely when $B, O$, $E, D$ all lie on a circle.
Prove that the product $C E \cdot E D$ attains its maximum, as $D$ varies, precisely when $B, O$, $E, D$ all lie on a circle.