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Let ABC be a triangle and let the bisectors of the angles at A, B, C meet the opposite sides at A', B', C'. Let

be the area of ABC and

be the area of A'B'C'. Let

and

be the lengths of BC, CA, and AB, respectively. Then

See problem 4.69 in V. Prasolov, Problems in Plane and Solid Geometry, Vol. 1, Plane Geometry [PDF], (D. Leites, ed. and trans.).

Contributed by: Jay Warendorff

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The Area of the Incentral Triangle