What is the value of $a+b$ where the area of the square in the diagram is $\dfrac{a}{b}$ and both are co-primes?

Question

The diagram shows two circles, each of radius $1$ and a square. The side length of the square can be written as $\dfrac{a}{b}$ ($a$ and $b$ are co-prime). Find $a+b$.

Source: Bangladesh Math Olympiad 2013 Junior Category.

The question only gives us the radius of the circle. Is this information enough? Is there any formula or way to get the area of the square from the radius of the circle only?

Answer 1

Let $A$ and $C$ be the centers of the circles, $B$ and $D$ the tangency points with the line. Denote furthermore the length of the square's side by $2k$.

Now extend $GH$ to meet $[AB]$ at $J$. By Pythagoras

$$[AJ]^2+[JH]^2=[AH]^2\iff (1-2k)^2+(1-k)^2=1$$

I think you can end it now...