[alogo] Areas-Ratio

Given the triangle DEF, construct triangle ABC as follows:
Take points G, H, I, on the sides of the triangle DEF, such that EG/EF = FH/FD = DI/DE = x (oriented segments). Then join G, H, I with D, E, F correspondingly to form triangle ABC. Show that the ratio of the areas y = a(ABC)/a(DEF) is related to x by the function y = (1-4*x+4*x^2)/(1-x+x^2)

Free movable in the figure below is:
The triangle DEF (switch to selection-tool (Ctrl+1) to catch and modify)
Movable-On-Line is:
The point G (switch to select-on-contour-tool (Ctrl+2) to catch and modify)

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]

Challenge: How could one generalize the problem to an arbitrary convex polygon?

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