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 $\sqrt{xy}.$
[asy] unitsize(1.5 cm); pair A, B, C, D, O, T; O = (0,0); T = dir(20); B = extension(T, T + rotate(90)*(T), (0,1), (1,1)); C = extension(T, T + rotate(90)*(T), (0,-1), (1,-1)); A = reflect((0,0),(0,1))*(B); D = reflect((0,0),(0,1))*(C); draw(A--B--C--D--cycle); draw(Circle(O,1)); label("$A$", A, NW); label("$B$", B, NE); label("$C$", C, SE); label("$D$", D, SW); [/asy]
