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This problem was very difficult for me.

 

1. 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 \(\sqrt{xy}.\)

 

Here's an image:

 

Thank you!

 Apr 11, 2020
 #1
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If x = y, then the quadrilateral is a square, and r = sqrt(xy) = x.  So it's true!

 Apr 11, 2020

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