(a(i))[2]
In Fig. $1$, $OAB$ is a sector of a circle with centre $O$ and radius $r$. $AX$ is tangent to the arc $AB$ at $A$, and $\angle BAX = \alpha$. Show that $\angle AOB = 2\alpha$.
(a(ii))[2]
Find the area of the shaded segment in terms of $r$ and $\alpha$.
(b)[6]
In Fig. $2$, $ABC$ is an equilateral triangle of side $4\,\text{cm}$. The lines $AX$, $BX$ and $CX$ are tangents to the equal circular arcs $AB$, $BC$ and $CA$. Using the results from part (a), determine the area of the shaded region, and give your answer in terms of $\pi$ and $\sqrt{3}$.