Recently I looked through an interesting book D. Wells "The Penquin book of Curious and Interesting Geometry" and came across this result, which I did not know about before: starting with a given quadrilateral , construct a square on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another. See the picture below:


It is interesting that this is true not only for a convex quadrilateral, but for arbitrary one and even self-intersecting one. This post is devoted to proving this result in Maple. The proof was very short and simple. Note that the coordinates of points are given not by lists, but by vectors. This is convenient when using  LinearAlgebra:-DotProduct  and  LinearAlgebra:-Norm  commands.

The code of the proof (the notation of the points on the picture coincide with their names in the code):

assign(seq(P||i=<x[i],y[i]>, i=1..4)):
assign(seq(Q||i=(P||i+P||(i+1))/2+<0,1; -1,0>.(P||(i+1)-P||i)/2, i=1..4)):
expand(DotProduct((Q||3-Q||1),(Q||4-Q||2), conjugate=false));
is(Norm(Q||3-Q||1, 2)=Norm(Q||4-Q||2, 2));

The output:


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