10.366 Two circles touch each other externally at point $A$. Find the radii of the circles if the chords connecting point $A$ with the points of tangency of one of their common external tangents are 6 and 8 cm.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-227.jpg?height=308&width=454&top_left_y=1151&top_left_x=191)

Let $BC$ be the common tangent to the two given circles $O_1$ and $O_2$ ($B$ and $C$ are the points of tangency), $AB = 8$, $AC = 6$. Draw the tangent at point $A$ until it intersects $BC$ at point $D$. $DA = DB = DC$ (Theorem 3).

$\Rightarrow \angle BAC = 90^\circ \Rightarrow BC = \sqrt{6^2 + 8^2} = 10, BD = 5$, $\angle AO_2C = 180^\circ - \angle ADC = \angle BDA \Rightarrow$

$\triangle ABD \sim \triangle ACO_2 \Rightarrow \frac{BD}{AO_2} = \frac{AB}{AC} \Rightarrow AO_2 = \frac{15}{4}$. Similarly, $AO_1 = \frac{20}{3}$.

Answer: $\frac{15}{4}$ cm, $\frac{20}{3}$ cm.