Quadrilateral $ABCD$ is an isosceles trapezoid, with bases $\overline{AB}$ and $\overline{CD}.$ A circle is inscribed in the trapezoid, as shown below. (In other words, the circle is tangent to all the sides of the trapezoid.) The length of base $\overline{AB}$ is $2x,$ and the length of base $\overline{CD}$ is $2y.$ Prove that the radius of the inscribed circle is $\sqrt{xy}.$

Currently I've labeled the center of the circle as O. Then i drew lines from O to the tangent points of the circle. The tangent point at the top is M, the bottom is N, the left point is S, and the right is T. I also drew lines from the center to the four corners of the trapezoid. I know that the triangles AOM and BOM are congruent, AOS and BOT are congruent, SOD and TOC are congruent, and DON and CON are congruent. But I don't know what to do now.

Can someone help me? Thanks!

Also please don't use trigonometry, whatever it is, I don't understand it.

Guest Sep 8, 2021