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Part of 1999 ITAMO

Problems(1)

AB contains intersection point of 2 tangent circles iff r_1 +r_2 = r

Source: 1999 ITAMO p3

1/25/2020
Let r1,r2,rr_1,r_2,r, with r1<r2<rr_1 < r_2 < r, be the radii of three circles Γ1,Γ2,Γ\Gamma_1,\Gamma_2,\Gamma, respectively. The circles Γ1,Γ2\Gamma_1,\Gamma_2 are internally tangent to Γ\Gamma at two distinct points A,BA,B and intersect in two distinct points. Prove that the segment ABAB contains an intersection point of Γ1\Gamma_1 and Γ2\Gamma_2 if and only if r1+r2=rr_1 +r_2 = r.
radiigeometrytangent circles