A circular arc centered at one vertex of a square of side length $4$ passes through two other vertices. A small circle is tangent to the large circle and to two sides of the square. What is the radius of the small circle?

Source: NCTM Mathematics Teacher 2006

SOLUTION

Triangle $ABC$ is a $45^\circ\!\textrm{-}45^\circ\!\textrm{-}90^\circ$ triangle of side length $r$
$AB=r\sqrt 2$
$AD$ is the diagonal of a square of side length $4$
$AD=4\sqrt 2$
$AD=AB+BD$
$4\sqrt 2=r\sqrt 2+(r+4)$
$4\sqrt 2-4=r\sqrt 2+r$
$4(\sqrt 2-1)=r(\sqrt 2+1)$
$r=\dfrac{4(\sqrt 2-1)}{\sqrt 2+1}$
Rationalize
$r=\dfrac{4(\sqrt 2-1)^2}{(\sqrt 2+1)(\sqrt 2-1)}$
$=\dfrac{4(2-2\sqrt 2+1)}{2-1}$
$=4(3-2\sqrt 2)$
$=12-8\sqrt 2$ unit

Answer: $12-8\sqrt 2$ unit