There are four circles. In the first one, a chord $A B$ is drawn, and the distance from the midpoint of the smaller of the two arcs formed to $A B$ is 1. The second, third, and fourth circles are located inside the larger segment and are tangent to the chord $A B$. The second and fourth circles are tangent to the inside of the first and externally to the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle, given that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.
Let $O_{1}, O_{2}, O_{3}$ and $O_{4}$ be the centers of the circles. Prove that $O_{1} O_{2} O_{3} O_{4}$ is a parallelogram.

## Solution

Denote by $O_{1}, O_{2}, O_{3}$ and $O_{4}$ the centers of the first, second, third, and fourth circles, respectively, and their radii as $r, x, y$, and $z$, respectively. By the condition, $x+y+z=r$. Since the first and second circles touch internally, and the third and fourth circles touch externally, then

$$
O_{1} O_{2}=r-x=y+z=O_{3} O_{4} \text {. }
$$

Similarly, we can prove that $O_{1} O_{4}=O_{2} O_{3}$.

Assume that points $O_{2}$ and $O_{4}$ lie on the same side of the line $O_{1} O_{3}$. Then $O_{2} O_{4} \| O_{1} O_{3}$, which contradicts the condition. Therefore, these points lie on opposite sides of the specified line, and $\mathrm{O}_{1} \mathrm{O}_{2} \mathrm{O}_{3} \mathrm{O}_{4}$ is a parallelogram.

Let $M$ be the midpoint of the smaller arc $A B$ of the first circle, $K$ the intersection of $O_{1} M$ and $A B$, $F$ the point of tangency of the fourth circle with the chord $A B$, $O_{2} P$ and $O_{3} Q$ the perpendiculars to $O_{1} M$ and $O_{4} F$, respectively. From the equality of the right triangles $O_{1} O_{2} P$ and $O_{4} O_{3} Q$, it follows that $O_{1} P=O_{4} Q$, or $r-1-x=z$ - $y$, from which

$$
y=x+z-r+1=(r-y)-r+1=1-y
$$

Therefore, $y=\frac{1}{2}$.

## Answer

$\frac{1}{2}$.

| 54796 Topics: | $\left[\begin{array}{l}\text { Tangent Circles } \\ \text { [Common Tangent to Two Circles] }\end{array}\right]$ |
| :---: | :---: |