Let a tetrahedron
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be inscribed in a sphere
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. Find the locus of points
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
inside the sphere
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
for which the equality
![\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4](/media/m/6/3/f/63f96e3ac4f709a5d1070611a829f66f.png)
holds, where
![A_1,B_1, C_1](/media/m/7/a/5/7a5631836183e8f85e1cd5453df8ad80.png)
, and
![D_1](/media/m/f/e/6/fe67388584f844e56a8db45e4e8768ca.png)
are the intersection points of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
with the lines
![AP,BP,CP](/media/m/e/3/5/e350e77de6b0f695c9f6b55675c46339.png)
, and
![DP](/media/m/f/d/8/fd8300e37d26cfa47fa724a9df058301.png)
, respectively.
%V0
Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality
$$\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4$$
holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.