*Grégoire Nicollier**March 29, 2016*

The following problem has been treated in http://www.cut-the-knot.org/m/Geometry/SquaresOnSidesOfQuadrilateral.shtml and http://www.cut-the-knot.org/m/Geometry/GalinkaGavrilenko.shtml.

We give here a generic proof of this kind of problem. The method applies to every linear circulant transformation of a planar n-gon (see for example the reference below, Sections 3-7). The proofs are then always very short. Napoleons's theorem, Van Aubel's theorem, the Petr-Douglas-Neumann theorem have for example such one-line proofs. It is also easy to discover "new" theorems in this way. We illustrate the method with the above problem.

We consider a quadrilateral Q of the complex plane as the point Q=(z_0,\,z_1,\,z_2,\,z_3) of \mathbf{C}^4 formed by the vertices in order. The dot product of two quadrilaterals Q and Q' (in this order) is

form the orthonormal *Fourier basis* of \mathbf{C}^4. In the Fourier basis, Q=(z_0,\,z_1,\,z_2,\,z_3) is thus given by

*spectrum*or

*discrete Fourier transform*of Q. Note that all Fourier basis polygons but F_0 are centered at the origin.

The *Fourier coefficient* \hat z_0 is the vertex centroid of Q. One has \hat z_2=0 if and only if z_0-z_1=z_3-z_2, that is, if and only if Q is a parallelogram. Each planar quadrilateral is hence the sum of a parallelogram P and a multiple of F_2, which translates the pairs of opposite vertices of P by opposite vectors.

Let T(Q) be the above quadrilateral M formed by the midpoints. The transformation T is linear. One sees graphically at once that

the transformation T is a filter that deletes the F_3-part of Q. Thus T(Q)-\hat z_0F_0 is the sum of the square 2\hat z_1F_1 and \hat z_2F_2: T(Q) has diagonals 4\hat z_1 and 4i\hat z_1. When Q is a parallelogram, one has \hat z_2=0 and T(Q)=\hat z_0F_0+2\hat z_1F_1 is a square.

Note that the Fourier basis vectors are eigenvectors of T: this is not by chance! The Fourier basis vectors are in fact eigenvectors of any circulant linear transformation of a polygon.

##### Reference

G. Nicollier, Some Theorems on Polygons With One-line Spectral Proofs,Forum Geom.15(2015) 267-273.

http://forumgeom.fau.edu/FG2015volume15/FG201527.pdf