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Line segment PQ is tangent to circle C

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September 15, 2010
geometrycircletangencyIsosceles TriangleIMO ShortlistIMO Longlist

Problem Statement

A circle CC with center OO on base BCBC of an isosceles triangle ABCABC is tangent to the equal sides AB,ACAB,AC. If point PP on ABAB and point QQ on ACAC are selected such that PB×CQ=(BC2)2PB \times CQ = (\frac{BC}{2})^2, prove that line segment PQPQ is tangent to circle CC, and prove the converse.
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