The Sum of the Square Roots of the Areas of Three Subtriangles

Let ABC be a triangle. Let A'A'', B'B'', and C'C'' be tangents to the incircle of ABC and parallel to BC, AC, and AB such that A'', B' are on AB, B'', C' are on BC, and C'', A' are on CA. Let be the area of ABC and , , and be the areas of AA'A'', BB'B'', and CC'C'' respectively. Then .



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