ABC is an equilateral triangle. The same color polygons are isometric.
I can prove that MPQR is a rectangle, but not that it's not a square.
MI = IP and RF' = F'Q = FN.
As MPQR is a rectangle MI = IP = RF' = F'Q = FN.
This equalities are independent of the choice of E and F.
PQ = QR is equivalent to : EM + EN = 2FN
Why in this case : PQ is not equal to RQ ?
Here is a solution to this problem.