Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\overline{AB}$, and the second is tangent to $\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? $\textbf{(A)}\ 18 \qquad$ $ \textbf{(B)}\ 27 \qquad$ $ \textbf{(C)}\ 36 \qquad$ $ \textbf{(D)}\ 81 \qquad$ $ \textbf{(E)}\ 108$
Solution
Assume $AB=2$. The first circle is the incircle of the equilateral triangle $GAB$ of side $2$, so its radius is $\frac{\sqrt3}3$.
So $GM=6, MN=MH=3$, $GH=9$. $PH=3\sqrt3$, so the ratio of the radii is $3\sqrt3:\frac{\sqrt3}3=9$, and the ratio of the areas is $9^2=81$.
Two circles lie outside the regular octagon $ABCDEFGH$. The first is tangent to $\overline{AB}$, and the second is tangent to $\overline{EF}$. Both are tangent to lines $BC$ and $HA$. What is the ratio of the radius of the second circle to the radius of the first circle?
