[Isosceles, Inscribed, and Circumscribed Trapezoids]

A circle with a radius of 5 is inscribed in some angle. The chord connecting the points of tangency is 8. Two tangents parallel to the chord are drawn to the circle. Find the sides of the resulting trapezoid.
See the solution to problem $\underline{52655}$.

## Solution

Let $O$ be the center of the circle, $P$ and $Q$ be the points of tangency of the circle with the lateral sides $CD$ and $AB$ of the resulting isosceles trapezoid $ABCD$, $K$ be the midpoint of $PQ$, and $M$ be the midpoint of $CD$.

From the similarity of triangles $PKO$ and $OPM$, we find that $OM = 25 / 4$. Since the midline of the isosceles circumscribed trapezoid is $25 / 2$, then

$CD = AB = 25 / 2$.

Let $N$ be the projection of vertex $C$ onto the larger base $AD$. From the similarity of triangles $CND$ and $PKO$, we find that $ND = 15 / 2$, and since the segment $AN$ is equal to the midline of the trapezoid, then $AD = AN + ND = 20$, $BC = AD - 2ND = 5$.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-13.jpg?height=417&width=463&top_left_y=1641&top_left_x=800)

Answer

$5 ; 20 ; 12.5 ; 12.5$.