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on each edge of a quadrilateral ABCD you build a square such that the points H, G, F, E are the centers of these squares (the intersection of the diagonals).

I need to use complex numbers to prove that FH and EG are equal and ortogonal to eachother.

Please help.

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I'm not sure how to start. – Ran Kashtan Nov 10 '12 at 12:38
up vote 0 down vote accepted

$1^{\rm st}$ hint:

Given two different complex numbers $a$ and $b$ the number $i(b-a)$ considered as a vector is $\vec{ab}$ rotated $90^\circ$ counterclockwise; in particular, it has the same length.

$2^{\rm nd}$ hint:

Assume that the quadrilateral is convex and that its vertices $A, B, C, D$ are arranged counterclockwise. Let $a, b, c, d$ be the complex coordinates of the four vertices. Then the other two vertices of the square over the side $AB$ have complex coordinates $a-i(b-a)$ and $b-i(b-a)$.

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I don't know how to use your hint :( – Ran Kashtan Nov 10 '12 at 12:38

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