## Small Circle

A circle is inscribed in a $60^\circ$ sector of a circle of radius $1$ (as shown). What is the radius of the small circle?

Source: NCTM Mathematics Teacher, February 2006

SOLUTION

$\overline{AC}$ is the angle bisector of $\angle A$ because $C$ is equidistant from $\overline{AB}$ and $\overline{AD}$. Triangle $ABC$ is a $30^\circ\!\textrm{-}60^\circ\!\textrm{-}90^\circ$ triangle with leg $BC=x$ and hypotenuse $AC=2x$.
$AC+CF=1$
$2x+x=1$
$x=1/3$
The radius of the small circle equals $1/3$.

Answer: $1/3$