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Ratio between areas of triangles defined by tangent circles

Source: South African MO 1995 Q4

May 27, 2012

ratiogeometrygeometry unsolved

Problem Statement

Three circles, with radii

p

q

and

r

and centres

A

B

and

C

respectively, touch one another externally at points

D

E

and

F

. Prove that the ratio of the areas of

\triangle DEF

and

\triangle ABC

equals

\frac{2pqr}{(p+q)(q+r)(r+p)}.