4. In $\triangle A B C$, $\angle A=45^{\circ}, \angle B=60^{\circ}$, then the ratio of the distances from the circumcenter $O$ to the three sides of $\triangle A B C$
Solution: Let the distances from $\mathrm{O}$ to the three sides $a, b, c$ be $\mathrm{r}_{a}, \mathrm{r}_{\mathrm{b}}, \mathrm{r}_{\mathrm{c}}$, respectively, then $\frac{r_{a} a}{r_{b} b}=\frac{S_{\triangle B O C}}{S_{\triangle A O C}}=\frac{\sin 2 A}{\sin 2 B}$, so $\frac{r_{a}}{r_{b}}=\frac{\cos A}{\cos B}$, similarly $\frac{r_{b}}{r_{c}}=\frac{\cos B}{\cos C}$, thus $\mathrm{r}_{a}: \mathrm{r}_{\mathrm{b}}: \mathrm{r}_{\mathrm{c}}=\cos \mathrm{A}: \operatorname{cosB}: \cos \mathrm{C}$ $=2: \sqrt{2}: \sqrt{3}-1$;