Difference between revisions of "2004 USAMO Problems/Problem 6"

Revision as of 18:46, 23 February 2012

A circle $\omega$ is inscribed in a quadrilateral $ABCD$ . Let $I$ be the center of $\omega$ . Suppose that

$(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2$ .

Prove that $ABCD$ is an isosceles trapezoid.

2004 USAMO (Problems • Resources)
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All USAMO Problems and Solutions

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 ==Problem==
-For what values of <math>k > 0 </math> is it possible to dissect a <math> 1 \times k </math> rectangle into two similar, but incongruent, polygons?
+A circle <math>\omega </math> is inscribed in a quadrilateral <math>ABCD </math>.  Let <math>I </math> be the center of <math>\omega </math>.  Suppose that
+<center>
+<math>
+(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2
+</math>.
+</center>
+Prove that <math>ABCD </math> is an [[isosceles trapezoid]].
 ==Solution==