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Lines in 3D space

Source: IMO LongList 1982 - P27

September 10, 2010
geometry3D geometrysphereIMO LonglistIMO Shortlist

Problem Statement

Let OO be a point of three-dimensional space and let l1,l2,l3l_1, l_2, l_3 be mutually perpendicular straight lines passing through OO. Let SS denote the sphere with center OO and radius RR, and for every point MM of SS, let SMS_M denote the sphere with center MM and radius RR. We denote by P1,P2,P3P_1, P_2, P_3 the intersection of SMS_M with the straight lines l1,l2,l3l_1, l_2, l_3, respectively, where we put Pi≠OP_i \neq O if lil_i meets SMS_M at two distinct points and Pi=OP_i = O otherwise (i=1,2,3i = 1, 2, 3). What is the set of centers of gravity of the (possibly degenerate) triangles P1P2P3P_1P_2P_3 as MM runs through the points of SS?
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