ILL 1982 Problem 27

Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ 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 $S_M$ denote the sphere with center $M$ and radius $R$ . We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$ , respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ( $i = 1, 2, 3$ ). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$ ?