ILL 1985 Problem 18

The circles $(R, r)$ and $(P, \rho)$ , where $r > \rho$ , touch externally at $A$ . Their direct common tangent touches $(R, r)$ at B and $(P, \rho)$ at $C$ . The line $RP$ meets the circle $(P, \rho)$ again at $D$ and the line $BC$ at $E$ . If $|BC| = 6|DE|$ , prove that:(a) the lengths of the sides of the triangle $RBE$ are in an arithmetic progression, and**(b)** $|AB| = 2|AC|.$