Six Incircles in an Equilateral Triangle

From an interior point P of an equilateral triangle ABC draw lines perpendicular to the sides. Let A', B', and C' be the points on the sides opposite A, B and C. Inscribe circles in the six subtriangles. Let (XYZ) denote the diameter of the incircle in triangle XYZ.

Then the sum of the diameters of the red circles equals the sum of the diameters of the blue circles:

(APC') + (BPA') + (CPB') = (C' PB) + (A' PC) + (B' PA)