Squares from any Quadrilateral

TL;DR

This paper generalizes a geometric construction involving rotations from triangles to quadrilaterals, showing that specific fixed points form parallelograms and that squares can be derived from any quadrilateral through algebraic methods.

Contribution

It extends Morley's theorem-inspired rotation fixed point results from triangles to quadrilaterals, revealing conditions under which squares emerge.

Findings

Fixed points of rotation products form six parallelograms for any quadrilateral.

When the quadrilateral is a parallelogram, certain parallelograms become squares.

A function is identified that produces squares from any quadrilateral when applied twice.

Abstract

In 1998 A. Connes proposed an algebraic proof of Morley's trisector theorem. He observed that the points of intersection of the trisectors are the fixed points of pairwise products of rotations around vertices of the triangle with angles two thirds of the corresponding angles of the triangle. This paper enquires for similar results when the initial polygon is an arbitrary quadrilateral. First we show that, when correctly gathered, fixed points of products of rotations around vertices of the quadrilateral with angles (2n+1)/2 of the corresponding angles of the quadrilateral form essentially six parallelograms for any integer n. Several congruence relations are exhibited between these parallelograms. Then, we show that if the original quadrilateral is itself a parallelogram, then for any integer n four of the resulting parallelograms are squares. Hence we present a function which, when applied twice, gives squares from any quadrilateral. The proofs use only algebraic methods of undergraduate level.