INMO 2016 Problem 5

Let $A B C$ be a right-angled triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $A C$ such that the in-radii of the triangles $A B D$ and $C B D$ are equal. If this common value is $r^{\prime}$ and if $r$ is the in-radius of triangle $A B C$, prove that

$$\frac{1}{r^{\prime}}=\frac{1}{r}+\frac{1}{B D}$$