
1. Let $A B C D$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B, D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A, C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $A C$ internally, the arc $B D$ internally and also touching the side $A B$. Find the radius of the circle $\Gamma$.


Solution: Let $O$ be the centre of $\Gamma$. By

![](https://cdn.mathpix.com/cropped/2024_06_05_bf025a87579e845b659fg-1.jpg?height=323&width=296&top_left_y=646&top_left_x=535)

symmetry $O$ is on the perpendicular bisector of $A B$. Draw $O E \perp A B$. Then $B E=A B / 2=1 / 2$. If $r$ is the radius of $\Gamma$, we see that $O B=1-r$, and $O E=r$. Using Pythagoras' theorem

$$
(1-r)^{2}=r^{2}+\left(\frac{1}{2}\right)^{2}
$$

Simplification gives $r=3 / 8$.
