OMCPLP 2021 Problem 5

There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$ . the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$ . There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines.

Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.