231.13.3.75 top vertex of a regular tetrahedron

[karush]

$\tiny{231.13.3.75}$
$\textrm{Imagine $3$ unit spheres
(radius equal to 1) with centers at,}\\$
$\textrm{$O(0,0,0)$, $P(\sqrt{3},-1,0)$ and $Q(\sqrt{3},1,0)$.} \\$
$\textrm{Now place another unit sphere symmetrically on top of these spheres with its center at R.} \\$
$\textrm{a Find the center of R.} \\$

 

[MarkFL]

The following post may give you some insight:

http://mathhelpboards.com/challenge-questions-puzzles-28/tetrahedral-stack-spheres-5676.html#post26011

 

[karush]

View attachment 7108

ok from this base

$\textrm{so if hieght of $h=\frac{\sqrt{3}}{2} a$ and $a=2$ then:}\\$
\begin{align*}\displaystyle
R&=\left(\sec \left(\frac{\pi }{6}\right),\frac{\sqrt{3}}{2} (2)\right)\\
&=\left( \frac{2\sqrt{3}}{3},\sqrt{3}\right)
\end{align*}

OK couldn't find some comprehensive equation for this so eyeballed it...