The Area of the Incentral Triangle
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.).