# Problem 5. (3 points)

In trapezoid $ABCD$, a point $X$ is taken on the base $AD$ such that segments $XB$ and $XC$ divide the trapezoid into three triangles that are similar to each other but pairwise unequal and non-isosceles. The side $AB$ has a length of 6. Find $AX \cdot DX$.
Answer: 36

## Solution:

The triangles are scalene, meaning all their angles are different.

$\alpha=\angle B X A=\angle X B C \neq \angle B X C=\beta$. The angle $\angle C X D$ together with these angles sums to $180^{\circ}$, so it is the third angle of the triangle and cannot be equal to either $\alpha$ or $\beta$.

If $\angle A B X=\beta$, we get equal angles around the common side in triangles $A B X$ and $C B X$, meaning these triangles are congruent. This contradicts the condition, so $\angle A B X=\gamma$, and accordingly, $\angle B A D=\beta$.

Similarly, we get $\angle X C D=\alpha$ and $\angle X D C=\beta$, from which the trapezoid is isosceles.

Now, let's write the ratio of the sides in similar triangles $A B X$ and $C D X$:

$$
\frac{D X}{C D}=\frac{A B}{A X}
$$

from which $D X \cdot A X=A B \cdot C D=36$.