[ Common tangent to two circles ]

The distance between the centers of two circles, one outside the other, is 65; the length of their common external tangent (between the points of tangency) is 63; the length of their common internal tangent is 25. Find the radii of the circles.
Drop a perpendicular from the center of one circle to the radius (or its extension) of the other circle, drawn to the point of tangency with the common external (or internal) tangent.

## Solution

Let $O_{1}$ and $O$ be the centers of the circles, $A$ and $B$ be the points of tangency with the common external tangent, $C$ and $D$ be the points of tangency with the internal tangent, and $r$ and $R (r < R)$ be the radii of the circles.

Drop perpendiculars $O_{1} K$ to $O B$ and $O P$ to the extension of $O_{1} C$. Then $(R-r)^{2}=O K^{2}=O_{1} O^{2}-O_{1} K^{2}=65^{2}-63^{2}$ $=128 \cdot 2=16^{2}$,

$(R+r)^{2}=O_{1} P^{2}=O_{1} O^{2}-O P^{2}=65^{2}-25^{2}=40 \cdot 90=60^{2}$.

From the system $R-r=16, R+r=60$, we find that $R=38$ and $r=22$.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-22.jpg?height=332&width=571&top_left_y=597&top_left_x=755)

## Answer

38 and 22.