31. In $\triangle A B C$ (see below), $A B=A C=\sqrt{3}$ and $D$ is a point on $B C$ such that $A D=1$. Find the value of $B D \cdot D C$.
31. Ans: 2 .
Construct a circle with $A$ as the center and $A B=A C=\sqrt{3}$ as the radius. Extend $A D$ to meet the circumference at $M$ and $N$ as shown.

Using the Intersecting Chord Theorem
$$
B D \cdot D C=M D \cdot N D=(\sqrt{3}-1)(\sqrt{3}+1)=2 .
$$