In a circle, intersecting chords $A B$ and $C D$ are perpendicular, $A D=m, B C=n$. Find the diameter of the circle.
Draw the diameter $C C_{1}$.

## Solution

Draw the diameter $C C_{1}$. Then the lines $C_{1} D$ and $A B$ are parallel, since both are perpendicular to the line $C D$. Therefore, $B C_{1}=A D=m$. From the right triangle $C B C_{1}$, we find that

$$
C C_{1}=\sqrt{B C^{2}+B C_{1}^{2}}=\sqrt{m^{2}+n^{2}}
$$

## Answer

$\sqrt{m^{2}+n^{2}}$.