[ Angles subtended by equal arcs and equal chords ] [ Properties and characteristics of the tangent $]$

In a circle, chords $A C$ and $B D$ intersect at point $E$, and the tangent to the circle passing through point $A$ is parallel to $B D$. It is known that $C D: E D=3: 2$ and $S_{\Delta \mathrm{ABE}}=8$. Find the area of triangle $A B C$.
Triangle $A C B$ is similar to triangle $A B E$, and triangle $A B E$ is similar to triangle $D C E$.

## Solution

The diameter of the circle passing through point $A$ is perpendicular to the given tangent, and therefore perpendicular to the chord $B D$, which means it bisects it. Thus, triangle $A B D$ is isosceles, so the arcs $A B$ and $A D$, not containing point $C$, are equal, hence,

$$
\angle A B E = \angle A B D = \angle A C D
$$

Triangles $A C B$ and $A B E$ are similar by two angles, and the similarity ratio is $\frac{A B}{A E}$. From the similarity of triangles $A B E$ and $D C E$, it follows that

$$
\frac{A B}{A E} = \frac{C D}{D E} = \frac{3}{2}
$$

Therefore,

$$
S_{\triangle \mathrm{ABC}} = \left(\frac{A B}{A E}\right)^{2} S_{\triangle \mathrm{ABE}} = \frac{9}{4} \cdot 8 = 18
$$

## Answer

18.