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USAMO 2000 Problem 2

Source:

October 1, 2005
AMCUSAMOgeometryinradiusquadratics

Problem Statement

Let SS be the set of all triangles ABCABC for which 5(1AP+1BQ+1CR)−3min⁡{AP,BQ,CR}=6r, 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, where rr is the inradius and P,Q,RP, Q, R are the points of tangency of the incircle with sides AB,BC,CA,AB, BC, CA, respectively. Prove that all triangles in SS are isosceles and similar to one another.
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