# Problem 5. (3 points)

In trapezoid $ABCD$, a point $X$ is taken on the base $BC$ such that segments $XA$ and $XD$ 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 5. Find $XC \cdot BX$.

Answer: 25
Solution:

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

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

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

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

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

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

from which $X C \cdot B X=A B \cdot C D=25$.