11.4. Let points $\mathrm{O}$ and $\mathrm{I}$ be the centers of the circumcircle and incircle of triangle $\mathrm{ABC}$, respectively. It is known that angle $\mathrm{AIO}$ is a right angle, and the measure of angle $\mathrm{CIO}$ is $45^{\circ}$. Find the ratio of the sides $\mathrm{AB:BC:CA}$.
Answer: 3:4:5

Solution. 1. The measure of angle AIC is $180^{\circ}-\frac{A+C}{2}=90^{\circ}+\frac{B}{2}>90^{\circ}$. If ray IO were outside angle AIC, the measure of angle AIO would equal the sum of the measures of angles AIC and CIO and would be greater than 90 degrees, which contradicts the condition. Therefore, ray IO lies inside angle AIC, so the measure of angle AIC equals the sum of the measures of angles AIO and CIO, that is, 135 degrees. This means that angle $\mathrm{ABC}$ is a right angle and triangle $\mathrm{ABC}$ is a right triangle with hypotenuse $\mathrm{AC}$, and point $\mathrm{O}$ is the midpoint of side AC.