6-0. Quadrilateral $A B C D$ is inscribed in a circle, and the areas of triangles $A B C$ and $A C D$ are equal. Three sides of the quadrilateral are 5, 8, and 10. Find all possible values of the length of the fourth side.
Answer: $4 ; 6.25 ; 16$

Solution. Since the quadrilateral is inscribed in a circle, its opposite angles $B$ and $D$ sum up to $180^{\circ}$. Therefore, their sines are equal. Then, from the equality of the areas of triangles $A B C$ and $A C D$, it follows that $A B \cdot B C=A D \cdot C D$, which gives three possible options, as the values of the three known sides can be arranged in three ways.