Problem

The lamina $BFDE$ is obtained by removing triangles $AED$ and $BCF$ from a uniform square lamina $ABCD$ of side $2a$. The length of side $AE$ is $a$ and the length of side $FC$ is $h$ (see diagram). The centre of mass of $BFDE$ is at a distance $\bar{x}$ from $AD$, and at a distance $\bar{y}$ from $AB$.

(a) Show that

$$\bar{x}=\frac{h^2-6ah+11a^2}{3(3a-h)}$$

and find a corresponding expression for $\bar{y}$.

(b) The lamina $BFDE$ is placed vertically on its edge $EB$ on a smooth horizontal surface.

Find, in terms of $a$, the set of possible values of $h$ for which the lamina remains in equilibrium.