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The lines intersect on the sphere

Source: Austrian Mathematical Olympiad 1999, Part 2, D1, P2

June 28, 2011
geometry3D geometryspheregeometric transformationconicsellipsegeometry unsolved

Problem Statement

Let ϵ\epsilon be a plane and k1,k2,k3k_1, k_2, k_3 be spheres on the same side of ϵ\epsilon. The spheres k1,k2,k3k_1, k_2, k_3 touch the plane at points T1,T2,T3T_1, T_2, T_3, respectively, and k2k_2 touches k1k_1 at S1S_1 and k3k_3 at S3S_3. Prove that the lines S1T1S_1T_1 and S3T3S_3T_3 intersect on the sphere k2k_2. Describe the locus of the intersection point.
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