
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 $A C$ externally, the arc $B D$ internally and also touching the side $A D$. Find the radius of the circle $\Gamma$.


Solution: Let $O$ be the centre of $\Gamma$ and $r$

![](https://cdn.mathpix.com/cropped/2024_06_05_2df523a359bae2f5e442g-1.jpg?height=328&width=312&top_left_y=644&top_left_x=519)
its radius. Draw $O P \perp A D$ and $O Q \perp A B$. Then $O P=r, O Q^{2}=O A^{2}-r^{2}=(1-r)^{2}-$ $r^{2}=1-2 r$. We also have $O B=1+r$ and $B Q=1-r$. Using Pythagoras' theorem we get

$$
(1+r)^{2}=(1-r)^{2}+1-2 r
$$

Simplification gives $r=1 / 6$.
