The Perpendicular Bisectors Construction, the Isoptic Point and the Simson Line of a Quadrilateral

TL;DR

This paper investigates an iterative process on quadrilaterals involving circumcenters, revealing a special limit point with properties akin to classical triangle centers and introducing the concept of a quadrilateral's Simson point and isogonal conjugation.

Contribution

It introduces a new iterative construction for quadrilaterals, characterizes the limit point with unique geometric properties, and extends classical triangle concepts to quadrilaterals.

Findings

The limit point is the common center of spiral similarities of triad circles.

The limit point has the isoptic property, visible from all triad circles at the same angle.

A unique Simson point exists with a pedal line forming the Simson line.

Abstract

Given a noncyclic quadrilateral, we consider an iterative procedure producing a new quadrilateral at each step. At each iteration, the vertices of the new quadrilateral are the circumcenters of the triad circles of the previous generation quadrilateral. The main goal of the paper is to prove a number of interesting properties of the limit point of this iterative process. We show that the limit point is the common center of spiral similarities taking any of the triad circles into another triad circle. As a consequence, the point has the isoptic property (i.e., all triad circles are visible from the limit point at the same angle). Furthermore, the limit point can be viewed as a generalization of a circumcenter. It also has properties similar to those of the isodynamic point of a triangle. We also characterize the limit point as the unique point for which the pedal quadrilateral is a parallelogram. Continuing to study the pedal properties with respect to a quadrilateral, we show that for every quadrilateral there is a unique point (which we call the Simson point) such that its pedal consists of four points on line, which we call the Simson line, in analogy to the case of a triangle. Finally, we define a version of isogonal conjugation for a quadrilateral and prove that the isogonal conjugate of the limit point is a parallelogram, while that of the Simson point is a degenerate quadrilateral whose vertices coincide at infinity.