Three equal radii tangential circles in equilateral triangle. What is the ratio of the $\color{blue}{blue}$ line to the $\color{red}{red}$ line?

Question

Consider the three tangential circles of equal radii inscribed in the equilateral triangle. What is the ratio of the $\color{blue}{blue}$ line to the $\color{red}{red}$ line?

The $\color{red}{\text{red line}}$ is simply the diameter of one of the circles.

The $\color{blue}{\text{blue line}}$ begins in the center of the top circle and ends on a line segment connecting the centers of the two bottom circles.

Answer 1

Let $\;O\;$ be the center of the circle with the red line and $\;M\;$ the one of the upper circle, with $\;A\;$ the point where these two lines intersect.

If the radius of the circles is $\;R\;$, then in the straight triangle $\;AOM\;$ we have

$$MA^2+AO^2=OM^2\implies MA+R^2=(2R)^2\implies MA=\sqrt3\,R$$

so

$$\frac{MA}{2R}=\frac{\sqrt3}2$$

Answer 2

Suppose the circles all have radius $1$ (and hence diameter $2$). Draw a line segment from the top of the blue line to the midpoint of the red line. You have a right triangle (along with the red and blue lines) with an angle of $60^\circ$; thus the blue line has length $\tan(60^\circ)=\sqrt3$.