[Pythagorean Theorem (direct and inverse)] $[\quad$ Chords and secants (other). $\quad]$

A circle of radius $r$ touches a certain line at point $M$. On this line, points $A$ and $B$ are taken on opposite sides of $M$, such that $M A = M B = a$.

Find the radius of the circle passing through points $A$ and $B$ and touching the given circle.
The line connecting the centers of two tangent circles passes through their point of tangency.

## Solution

Let $R$ be the radius of the desired circle, $O_{1}$ its center, $K$ the point of tangency of the circles (internal tangency), and $O$ the center of the given circle.

Since triangle $M B O_{1}$ is a right triangle, we have $R^{2}=(R-2 r)^{2}+a^{2}$, from which $R=\frac{a^{2}+4 r^{2}}{4 r}$.
![](https://cdn.mathpix.com/cropped/2024_05_06_c271881b12104a3b803eg-23.jpg?height=450&width=1144&top_left_y=2360&top_left_x=457)

## Answer

## Problem