Let \(ABCD\) be 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 \(\sqrt{xy}.\)
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