IMO 1982 SL 18

Problem

C6 (FRA 2) Let O be a point of three-dimensional space and let l1, l2, l3 be mutually perpendicular straight lines passing through O. Let S denote the sphere with center O and radius R, and for every point M of S, let SM denote the sphere with center M and radius R. We denote by P1, P2, P3 the intersection of SM with the straight lines l1, l2, l3, respectively, where we put Pi ̸= O if li meets SM at two distinct points and Pi = O otherwise (i = 1, 2, 3). What is the set of centers of gravity of the (possibly degenerate) triangles P1P2P3 as M runs through the points of S?

Solution

Set the coordinate system with the axes x, y, z along the lines l1, l2, l3 respectively. The coordinates (a, b, c) of M satisfy a2 + b2 + c2 = R2, and so SM is given by the equation (x−a)2+(y−b)2+(z−c)2 = R2. Hence the coordinates of P1 are (x, 0, 0) with (x −a)2 + b2 + c2 = R2, implying that either x = 2a or x = 0. Thus by the definition we obtain x = 2a. Similarly, the coordinates of P2 and P3 are (0, 2b, 0) and (0, 0, 2c) respectively. Now, the centroid of \triangleP1P2P3 has the coordinates (2a/3, 2b/3, 2c/3). Therefore the required locus of points is the sphere with center O and radius 2R/3.