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sphere S remains tangent to a fixed sphere

Source: Romania IMO TST 1989 2.4

February 17, 2020
spherefixedtangent spheres3D geometrytrihedral anglegeometry

Problem Statement

Let A,B,CA,B,C be variable points on edges OX,OY,OZOX,OY,OZ of a trihedral angle OXYZOXYZ, respectively. Let OA=a,OB=b,OC=cOA = a, OB = b, OC = c and RR be the radius of the circumsphere SS of OABCOABC. Prove that if points A,B,CA,B,C vary so that a+b+c=R+la+b+c = R+l, then the sphere SS remains tangent to a fixed sphere.
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