Given that three congruent circles of radius $4$ and one large circle of unknown radius are inscribed in equilateral triangle $ABC$, and all of the circles are tangent to each other and the triangle.
1) What is the side length of the equilateral triangle?
2) What is the radius of circle $O$ in the figure below?
Some context:
I have tried to construct a 30-60-90 triangle from the tangent of circle $R$ to find that $RC$ = 8 and by similarity finding the lengths of triangle $COX$ (where $X$ is the point of tangency of circle $O$ on the side $BC$ of triangle $ABC$) but I cannot seem to find the radius of circle $O$ or the length of $CX$.
I know that the centroid of an equilateral triangle is always the center (this is also the incenter and the circumcenter), and the height of an equilateral triangle is $\frac{s{\sqrt 3}}{2}$ and the area is $\frac{{\sqrt 3}}{4}{s^2}$.
I figured that finding the side length of the equilateral triangle would involve finding the altitude that passes through points $O$, $R$, and $C$, and that the radius of circle $R$ + the diameter of circle $O$ + $RC$ but I didn't have a good method to find the diameter of circle $O$.
I also understand that circles $P$, $R$, and $Q$ form circumscribed angles at vertices $A$, $C$, and $B$, respectively, but I was unsure of how that helped to solve the problem.
Best Answer
Circle with the center $O$ is inscribed in the equilateral triangle $ABC$ with the height $CF$ and inradius $OF$, circle with the center $R$ is inscribed in the equilateral triangle $DEC$ with the height $CG$ and inradius $RG$,
\begin{align} |CF|&=3|CG| , \end{align}
hence, the radius
\begin{align} |OF|&=3\cdot|RG|=3\cdot4=12 , \end{align}
and the side length of $\triangle ABC$ \begin{align} |AC|&= \frac{|CF|}{\tfrac {\sqrt3}2} =\frac{3\cdot|OF|}{\tfrac {\sqrt3}2} =2\sqrt3\cdot|OF|=24\sqrt3 . \end{align}