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Two circles

Source: Tournament of Towns, Fall 2002, Senior A Level, P5

May 17, 2014
projective geometrygeometrycyclic quadrilateralgeometry proposed

Problem Statement

Two circles Γ1,Γ2\Gamma_1,\Gamma_2 intersect at A,BA,B. Through BB a straight line ℓ\ell is drawn and ℓ∩Γ1=K,ℓ∩Γ2=M  (K,M≠B)\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B). We are given ℓ1∥AM\ell_1\parallel AM is tangent to Γ1\Gamma_1 at QQ. QA∩Γ2=R  (≠A)QA\cap \Gamma_2=R\;(\neq A) and further ℓ2\ell_2 is tangent to Γ2\Gamma_2 at RR. Prove that:
[*]ℓ2∥AK\ell_2\parallel AK [*]ℓ,ℓ1,ℓ2\ell,\ell_1,\ell_2 have a common point.
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