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 \( P_{i} \neq O \) 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 \( P_{1} P_{2} P_{3} \) as \( M \) runs through the points of \( S \) ?
18. Set the coordinate system with the axes $x, y, z$ along the lines $l_{1}, l_{2}, l_{3}$ respectively. The coordinates $(a, b, c)$ of $M$ satisfy $a^{2}+b^{2}+c^{2}=R^{2}$, and so $S_{M}$ is given by the equation $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=R^{2}$. Hence the coordinates of $P_{1}$ are $(x, 0,0)$ with $(x-a)^{2}+b^{2}+c^{2}=R^{2}$, implying that either $x=2 a$ or $x=0$. Thus by the definition we obtain $x=2 a$. Similarly, the coordinates of $P_{2}$ and $P_{3}$ are $(0,2 b, 0)$ and $(0,0,2 c)$ respectively. Now, the centroid of $\triangle P_{1} P_{2} P_{3}$ has the coordinates $(2 a / 3,2 b / 3,2 c / 3)$. Therefore the required locus of points is the sphere with center $O$ and radius $2 R / 3$.