On the inside of the triangle (0,0),(1,0),(0,1), define the "pretty useless map" or pump by the following prescription:
Given a point A find its projections x0,y0 to the x and y axes and also x1,y1 to the same axes shifted by one unit. Connect x1 to y0 and y1 to x0. The intersection of these two straight lines shall be pump(A)
Given a triangle ABC show that the area of triangle pump(A)pump(B)pump(C) can be written as f(X) for some fixed scalar function f where X is the area of ABC.
Most elegant solution wins.
A purely geometric, zero algebra solution exists.
Hint 2 (Warning: this really is a spoiler):