Euclidean Geometry proof

Question

In triangle $ABC$, $AB = AC$ and $A-D-B$ such that $DC = BC$. Prove: $$BC =\sqrt{(AB*BD)}$$.

I've tried to use altitudes and the Pythagorean theorem but I'm completely lost on how to solve this problem.

Answer 1

Hint: The triangles $ABC$ and $BCD$ are similar because they are both isosceles and share a base angle.