In parallelogram $A B C D$, diagonal $B D$ is equal to 2, angle $C$ is $45^{\circ}$, and line $C D$ is tangent to the circumcircle of triangle $A B D$. Find the area of parallelogram $A B C D$.
Prove that $B D \perp B C$.

## Solution

It is clear that $\angle A=\angle C=45^{\circ}$. Angle $\angle B A D$ is inscribed in the circle, and $\angle B D C$ is the angle between the tangent and the chord $B D$, therefore,

$$
\angle B D C=\angle A=45^{\circ}
$$

Thus, $\angle C B D=90^{\circ}$. Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_16ab69fab1743849dba9g-16.jpg?height=429&width=602&top_left_y=734&top_left_x=731)

## Answer

4.