Let P be a point in the interior of an equilateral triangle.

Three line segments connect P with the vertices of the 
triangle and three line segments connect P perpendicularly
to the sides of the triangle.

These six line segments divide the triangle into six smaller
triangles that surround P.

If u, v, w, x, y, and z denote the areas of the triangles 
around P in that order, then prove that

                     u + w + y = v + x + z.

An immediate fact from my grey matter is that regardless of where P is, the sum of its distances from the edges is constant. (Based on conservation of the area of the given equilateral triangle.)

Is the solution along this line?