[ Properties and characteristics of the tangent ] $[$ Diameter, main properties $]$

On one side of a right angle with vertex at point $O$, two points $A$ and $B$ are taken, such that $O A=a, O B=b$. Find the radius of the circle passing through points $A$ and $B$ and tangent to the other side of the angle.
Drop a perpendicular from the center of the circle to the chord $A B$.

## Solution

Assume that $b>a$. Let $M$ be the point of tangency, $R$ be the desired radius, and $K$ be the projection of the center $O_{1}$ of the given circle onto $A B$. Then

$$
\begin{aligned}
A K=K B= & \frac{b-a}{2}, R=M O_{1}=O K=O A+A K= \\
& =a+\frac{b-a}{2}=\frac{b+a}{2}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-20.jpg?height=277&width=306&top_left_y=-1&top_left_x=882)

## Answer

$\frac{a+b}{2}$.