Question 18
C6 (FRA 2). 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 $P_i\ne O$ 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 $S$?