IR-MO 2002 Problem 6

$M$ is midpoint of $BC$ . $P$ is an arbitary point on $BC$ . $C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$ Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P. $C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$ . Prove : $$r_{1}+r_{2}+r_{3}=R$$ ( $R$ radius of big circle)