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problem 5 of Indian Mathematical Olympiad 2008

Source: Plane Geometry (line joining in-centre and circum-centre of a triangle)

September 1, 2009
geometrycircumcircleincentergeometric transformationhomothetyangle bisector

Problem Statement

Let ABC ABC be a triangle; ΓA,ΓB,ΓC \Gamma_A,\Gamma_B,\Gamma_C be three equal, disjoint circles inside ABC ABC such that ΓA \Gamma_A touches AB AB and AC AC; ΓB \Gamma_B touches AB AB and BC BC; and ΓC \Gamma_C touches BC BC and CA CA. Let Γ \Gamma be a circle touching circles ΓA,ΓB,ΓC \Gamma_A, \Gamma_B, \Gamma_C externally. Prove that the line joining the circum-centre O O and the in-centre I I of triangle ABC ABC passes through the centre of Γ \Gamma.
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