TITLE: What is the radius of this circle?
QUESTION [9 upvotes]: Polygon $ABC$ is an equilateral triangle. Three congruent semicircular arcs are drawn on the three sides and towards the interior of the triangle. The terminal points of the semicircular arcs are the three vertices of the triangle $ABC$.
Semicircle $C_1$ of diameter $AB$
Semicircle $C_2$ of diameter $BC$
Semicircle $C_3$ of diameter $CA$
A small circle $F$ is drawn such that it is tangent internally to $C_1$ and $C_3$ and externally to $C_2$
What is the radius of circle $ F$?
Here is the picture

REPLY [4 votes]: From the red triangle in the picture we infer
$$(1-r)^2=\left({1\over2}\right)^2+\left(r+\left(1-{\sqrt{3}\over2}\right)\right)^2\ ,$$
and this leads to
$$r={3\sqrt{3}-1\over13}\ .$$