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Four circles

Source: Canada Mathematical Olympiad 2007

July 26, 2007
geometrycircumcirclegeometric transformationhomothetysymmetrycyclic quadrilateralHi

Problem Statement

Let the incircle of triangle ABC ABC touch sides BC, CA BC,\, CA and AB AB at D, E D,\, E and F, F, respectively. Let ω, ω1, ω2 \omega,\,\omega_{1},\,\omega_{2} and ω3 \omega_{3} denote the circumcircles of triangle ABC, AEF, BDF ABC,\, AEF,\, BDF and CDE CDE respectively. Let ω \omega and ω1 \omega_{1} intersect at A A and P, ω P,\,\omega and ω2 \omega_{2} intersect at B B and Q, ω Q,\,\omega and ω3 \omega_{3} intersect at C C and R. R. a. a. Prove that ω1, ω2 \omega_{1},\,\omega_{2} and ω3 \omega_{3} intersect in a common point. b. b. Show that PD, QE PD,\, QE and RF RF are concurrent.
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