From an external point $A$, a tangent $A B$ and a secant $A C D$ are drawn to a circle. Find the area of triangle $C B D$, if $A C: A B=2: 3$ and the area of triangle $A B C$ is 20.
The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.

## Solution

By the theorem on the angle between a tangent and a chord, $\angle A B C=\angle A D B$. Therefore, triangle $A D B$ is similar to triangle $A B C$ (by two angles). The similarity coefficient is the ratio of the corresponding sides, i.e., $\frac{A B}{A C} = \frac{3}{2}$. Therefore,

\[
\begin{aligned}
& S_{\triangle \mathrm{ADB}}=\left(\frac{3}{2}\right)^{2} S_{\triangle \mathrm{ABC}}=\frac{9}{4} \cdot 20=45 \\
& S_{\triangle \mathrm{CBD}}=S_{\triangle \mathrm{ADB}}-S_{\triangle \mathrm{ABC}}=45-20=25
\end{aligned}
\]

![](https://cdn.mathpix.com/cropped/2024_05_06_5ddd4eab4fe1a12edeb9g-04.jpg?height=391&width=840&top_left_y=-1&top_left_x=615)

## Answer

25.