Parabola

Let $\mathrm{A}(0,1), \mathrm{B}(1,1)$ and $\mathrm{C}(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $\mathrm{D}$. If the focus of the parabola $y^{2}=4 \mathrm{ax}$ passing through $\mathrm{D}$ is $(\alpha+\beta \sqrt{2}, 0)$, where $\alpha$ and $\beta$ are rational numbers, then $\frac{\alpha}{\beta^{2}}$ is equal to :