[ Tangent lines and tangent circles ] Described quadrilaterals $\quad]$

Two circles of radii 4 and 3 touch each other externally. Common external tangents $P Q$ and $R S$ are drawn to these circles such that points $P$ and $S$ lie on the circle of the larger radius, and points $Q$ and $R$ lie on the circle of the smaller radius. Find the radius of the circle that touches segments $R S$, $S P$, and $P Q$.
From the theorem about the angle between a tangent and a chord, it follows that the point of tangency of the circles is the point of intersection of the angle bisectors of the given quadrilateral, i.e., the center of its inscribed circle. It remains to find the distance from this point to the common external tangent of the given circles.

![](https://cdn.mathpix.com/cropped/2024_05_06_ae1b0d3bbbf9aa5c38e8g-33.jpg?height=528&width=866&top_left_y=-1&top_left_x=593)

## Answer

$\frac{24}{7}$.