A Pre-RMO question! -24

Geometry Level 2

In rectangle $ABCD$ , $\overline{AB}=8$ and $\overline{BC} = 20$ . Let $P$ be a point on $\overline{AD}$ such that $\angle BPC = 90\degree$ .

If $r_1, r_2, r_3$ are the radii of the in-circles of triangles $\triangle {APB}, \triangle {BPC}$ and $\triangle CPD$ .

What is the value of $r_1 + r_2 + r_3$ ?