Question 3
Let $(O)$ be a circle with diameter $AB$. A variable point $P\ne B$ lies on the tangent at $B$. Line $PA$ meets $(O)$ again at $C$. Let $D$ be the reflection of $C$ in $O$, and let $PD$ meet $(O)$ again at $E$.
1. Prove that $AE,BC,PO$ pass through a common point $M$.
2. Determine the positions of $P$ for which the area of triangle $AMB$ is greatest, and find that maximum in terms of the radius $R$.
1. Prove that $AE,BC,PO$ pass through a common point $M$.
2. Determine the positions of $P$ for which the area of triangle $AMB$ is greatest, and find that maximum in terms of the radius $R$.