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O_1P, O_2Q tangents to (O_1), (O_2) [INMO 2013 P1]

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February 3, 2013
geometrycircumcircle

Problem Statement

Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles touching each other externally at R.R. Let O1O_1 and O2O_2 be the centres of Γ1\Gamma_1 and Γ2,\Gamma_2, respectively. Let ℓ1\ell_1 be a line which is tangent to Γ2\Gamma_2 at PP and passing through O1,O_1, and let ℓ2\ell_2 be the line tangent to Γ1\Gamma_1 at QQ and passing through O2.O_2. Let K=ℓ1∩ℓ2.K=\ell_1\cap \ell_2. If KP=KQKP=KQ then prove that the triangle PQRPQR is equilateral.
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