[ Sine Theorem

The bases of an isosceles trapezoid are in the ratio 5:12, and its height is 17. Find the radius of the circle circumscribed around the trapezoid, given that its midline is equal to the height.

## Answer

13. 

Apply the formula $a=2 R \sin \boldsymbol{\alpha}$.

The diagonal $A C$ of the inscribed quadrilateral $A B C D$ is the bisector of angle $D A B$.

Prove that one of the two triangles cut off from triangle $A B C$ by the diagonal $B D$ is similar to triangle $A B C$.

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Let $K$ be the point of intersection of the diagonals of a quadrilateral. Prove that triangle $B K C$ is similar to triangle $A B C$.

## Solution

Let $K$ be the point of intersection of the diagonals $A C$ and $B D$. Since $\angle K B C = \angle D B C = \angle D A C = \angle B A C$, triangle $B K C$ is similar to triangle $A B C$ by two angles (the angle at vertex $C$ is common).

![](https://cdn.mathpix.com/cropped/2024_05_06_5ddd4eab4fe1a12edeb9g-18.jpg?height=478&width=504&top_left_y=790&top_left_x=777)