Let
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
be a point of three-dimensional space and let
![l_1, l_2, l_3](/media/m/5/8/9/589bd14418838013f437e552b9cdd560.png)
be mutually perpendicular straight lines passing through
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
. Let
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
denote the sphere with center
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
and radius
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
, and for every point
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
, let
![S_M](/media/m/e/f/6/ef66189d71bb23040b4d5d15e37f498d.png)
denote the sphere with center
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and radius
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
. We denote by
![P_1, P_2, P_3](/media/m/0/9/b/09b69a1e26a9b8213563a1a2f1d307c1.png)
the intersection of
![S_M](/media/m/e/f/6/ef66189d71bb23040b4d5d15e37f498d.png)
with the straight lines
![l_1, l_2, l_3](/media/m/5/8/9/589bd14418838013f437e552b9cdd560.png)
, respectively, where we put
![P_i \neq O](/media/m/e/a/4/ea4f316ece95f62da0d948dfa4e5f43b.png)
if
![l_i](/media/m/b/2/6/b2647bded77d01395618ce39635fe411.png)
meets
![S_M](/media/m/e/f/6/ef66189d71bb23040b4d5d15e37f498d.png)
at two distinct points and
![P_i = O](/media/m/c/8/7/c87a1442d45d41b4fbcf77362b0581f5.png)
otherwise (
![i = 1, 2, 3](/media/m/a/7/b/a7bbda360cece7bb6d0d6a1f43153f17.png)
). What is the set of centers of gravity of the (possibly degenerate) triangles
![P_1P_2P_3](/media/m/3/5/5/3556b2178ef8de9f97d48162d503e6c2.png)
as
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
runs through the points of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
?
%V0
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 $S$?