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IMO LongList 1985 CYP1 - Tangent and Circles

Source:

September 10, 2010
arithmetic sequencegeometry

Problem Statement

The circles (R,r)(R, r) and (P,ρ)(P, \rho), where r>ρr > \rho, touch externally at AA. Their direct common tangent touches (R,r)(R, r) at B and (P,ρ)(P, \rho) at CC. The line RPRP meets the circle (P,ρ)(P, \rho) again at DD and the line BCBC at EE. If ∣BC∣=6∣DE∣|BC| = 6|DE|, prove that: (a) the lengths of the sides of the triangle RBERBE are in an arithmetic progression, and
(b) ∣AB∣=2∣AC∣.|AB| = 2|AC|.
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