
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 arcs $A C$ and $B D$ both externally and also touching the side $C D$. Find the radius of the circle $\Gamma$.


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

![](https://cdn.mathpix.com/cropped/2024_06_05_c2bde1f57585502709b7g-1.jpg?height=323&width=311&top_left_y=646&top_left_x=533)
metry $O$ is on the perpendicular bisector of $C D$. Draw $O L \perp C D$ and $O K \perp B C$. Then $O K=C L=C D / 2=1 / 2$. If $r$ is the radius of $\Gamma$, we see that $B K=1-r$, and $O E=r$. Using Pythagoras' theorem

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

Simplification gives $r=1 / 16$.
