21. In a circle $\omega$ centred at $O, A A^{\prime}$ and $B B^{\prime}$ are diameters perpendicular to each other such that the points $A, B, A^{\prime}, B^{\prime}$ are arranged in an anticlockwise sense in this order. Let $P$ be a point on the minor arc $A^{\prime} B^{\prime}$ such that $A P$ intersects $B B^{\prime}$ at $D$ and $B P$ intersects $A A^{r}$ at $C$. Suppose the area of the quadrilateral $A B C D$ is 100 . Find the radius of $\omega$.
21. Answer: 10
Solution. Since $A C$ intersects $B D$ at right angle, the area of the convex quadrilateral $A B C D$ is $\frac{1}{2} A C \cdot B D$. Let $M$ be the midpoint of $P B$. As $\angle C A B=\angle A B D=45^{\circ}$, and $\angle B C A=\angle B O M=\angle D A B$, we have $\triangle A B C$ is similar to $\triangle B D A$. Thus $A B / B D=$ $A C / B A$. From this, we have $(A B C D)=\frac{1}{2} A C \cdot B D=\frac{1}{2} A B^{2}=O A^{2}$ so that $O A=10$.