Finding all possible proofs

I find the following quite instructive:

Obviously \begin{align} a&=r+y \\ &=r+\sqrt{r^2 - \left(\frac{a}{2}\right)^2} \end{align} and solving gives $$\frac{r}{a} = \frac{5}{8} \, .$$

Let $I$ be the midpoint of $[AB]$. Use the Pythagorean theorem in triangle $AIM$: $$r^2=\dfrac{a^2}{4}+(a-r)^2$$ to deduce $a=8r/5$.