Let

be a point of three-dimensional space and let

be mutually perpendicular straight lines passing through

. Let

denote the sphere with center

and radius

, and for every point

of

, let

denote the sphere with center

and radius

. We denote by

the intersection of

with the straight lines

, respectively, where we put

if

meets

at two distinct points and

otherwise (

). What is the set of centers of gravity of the (possibly degenerate) triangles

as

runs through the points of

?
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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 $S$?