## Three Tangent Circles

Three mutually tangent circles of radius one are surrounded by a larger circle that is simultaneously tangent to all three. What is the radius of the larger circle?
Source: www.mathcircles.org

SOLUTION

The centers $A,B,C$ of the three mutually tangent circles form an equilateral triangle $ABC$ with side = 2. The altitudes are also medians and angle bisectors (not all are drawn for clarity). They all intersect at point $O$.
Length of altitude:
$AH=\sqrt{3}$
Ratio of median:
$OA:AH=2:3$

$\frac{OA}{AH}=\frac{2}{3}$
$\frac{OA}{\sqrt{3}}=\frac{2}{3}$
$OA=\frac{2\sqrt{3}}{3}$

$OP=OA+AP$
$=\frac{2\sqrt{3}}{3}+1$
$=\frac{2\sqrt{3}+3}{3}$
Answer: $\frac{2\sqrt{3}+3}{3}$