NZ-NZMO 2023 Problem 6

Let triangle $ABC$ be right-angled at $A$ . Let $D$ be the point on $AC$ such that $BD$ bisects angle $\angle ABC$ . Prove that $BC - BD = 2AB$ if and only if $\frac{1}{BD} - \frac{1}{BC} = \frac{1}{2AB}$ .