6. [5] Let $A B C$ be a triangle with $\angle A=45^{\circ}$. Let $P$ be a point on side $B C$ with $P B=3$ and $P C=5$. Let $O$ be the circumcenter of $A B C$. Determine the length $O P$.
Answer: $\sqrt{17}$ Using extended Sine law, we find the circumradius of $A B C$ to be $R=\frac{B C}{2 \sin A}=4 \sqrt{2}$. By considering the power of point $P$, we find that $R^{2}-O P^{2}=P B \cdot P C=15$. So $O P=\sqrt{R^{2}-15}=$ $\sqrt{16 \cdot 2-15}=\sqrt{17}$