MathDB
Isosceles triangle with inscribed semicircle

Source: Cono Sur 2008 #5

November 17, 2015
geometrycono sur

Problem Statement

Let ABCABC be an isosceles triangle with base ABAB. A semicircle Γ\Gamma is constructed with its center on the segment AB and which is tangent to the two legs, ACAC and BCBC. Consider a line tangent to Γ\Gamma which cuts the segments ACAC and BCBC at DD and EE, respectively. The line perpendicular to ACAC at DD and the line perpendicular to BCBC at EE intersect each other at PP. Let QQ be the foot of the perpendicular from PP to ABAB. Show that PQCP=12ABAC\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}.
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