MathDB
identity

Source: Ireland 2007

July 6, 2009
geometryinequalitiesgeometry proposed

Problem Statement

Let ABC ABC be a triangle the lengths of whose sides BC,CA,AB, BC,CA,AB, respectively, are denoted by a,b, a,b, and c c. Let the internal bisectors of the angles ∠BAC,∠ABC,∠BCA, \angle BAC, \angle ABC, \angle BCA, respectively, meet the sides BC,CA, BC,CA, and AB AB at D,E, D,E, and F F. Denote the lengths of the line segments AD,BE,CF AD,BE,CF by d,e, d,e, and f f, respectively. Prove that: def\equal{}\frac{4abc(a\plus{}b\plus{}c) \Delta}{(a\plus{}b)(b\plus{}c)(c\plus{}a)} where Δ \Delta stands for the area of the triangle ABC ABC.
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