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 \(\sqrt{xy}.\)
HINT: When you have a circle that is tangent to lines, connect the center of the circle to the points of tangency.