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AO/OD - BC/AT=4

Source: Turkey TST 2013 - Day 1 - P3

April 2, 2013
geometrycircumcircleincentergeometric transformationreflectionparallelograminradius

Problem Statement

Let OO be the circumcenter and II be the incenter of an acute triangle ABCABC with m(B^)≠m(C^)m(\widehat{B}) \neq m(\widehat{C}). Let DD, EE, FF be the midpoints of the sides [BC][BC], [CA][CA], [AB][AB], respectively. Let TT be the foot of perpendicular from II to [AB][AB]. Let PP be the circumcenter of the triangle DEFDEF and QQ be the midpoint of [OI][OI]. If AA, PP, QQ are collinear, prove that ∣AO∣∣OD∣−∣BC∣∣AT∣=4.\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.
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