[Paper Review] On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas

Emmanuel Tsukerman|arXiv (Cornell University)|Dec 31, 2011

Mathematics and Applications3 references3 citations

TL;DR

This paper establishes equidistant Simson polygons as discrete analogs of parabolas, demonstrating that they converge to a parabola in the limit and inherit key geometric properties such as the optical reflection property, Archimedes' Lemma, and Lambert’s Theorem. The core contribution is a rigorous discrete-geometric framework linking Simson polygons to parabolic geometry via a limit process.

ABSTRACT

We begin by proving a few general facts about Simson polygons, defined as polygons which admit a pedal line. We use an inductive argument to show that no convex $n$-gon, $n\geq5$, admits a Simson Line. We then determine a property which characterizes Simson $n$-gons and show that one can be constructed for every $n\geq3$. We proceed to show that a parabola can be viewed as a limit of special Simson polygons, called equidistant Simson polygons, and that these polygons provide the best piecewise linear continuous approximation to the parabola. Finally, we show that equidistant Simson polygons can be viewed as discrete analogs of parabolas and that they satisfy a number of results analogous to the pedal property, optical property, properties of Archimedes triangles and Lambert's Theorem of parabolas. The corresponding results for parabolas are easily obtained by applying a limit process to the equidistant Simson polygons.

Motivation & Objective

• To investigate whether n-gons with n ≥ 5 can admit a Simson line, extending the Simson-Wallace Theorem beyond triangles and quadrilaterals.
• To characterize Simson polygons and construct them for all n ≥ 3.
• To show that equidistant Simson polygons converge to a parabola and provide the best piecewise linear approximation to it.
• To establish discrete analogs of classical parabolic theorems—optical property, Archimedes’ Lemma, and Lambert’s Theorem—via these polygons.
• To unify discrete polygonal geometry with continuous parabolic geometry through a limit process.

Proposed method

• Using an inductive argument to prove that no convex n-gon with n ≥ 5 admits a Simson point, relying on cyclic quadrilateral properties and Miquel point geometry.
• Defining equidistant Simson polygons via symmetric placement of vertices along a parabola-like path with constant Δ in x-coordinate.
• Applying coordinate geometry to show that midpoints of sides of equidistant Simson polygons lie on a parabola y = x²/(4s) with focus S.
• Using reflection and orthogonality properties to prove that reflections of lines orthogonal to the Simson line through polygon sides pass through the focus S.
• Leveraging the Simson-Wallace Theorem to prove that the circumcircle of any triangle formed by three sides of a Simson polygon passes through the Simson point.
• Taking the limit as n → ∞ and Δ → 0 to recover classical parabolic theorems from discrete polygonal counterparts.

Experimental results

Research questions

• RQ1 Can convex n-gons with n ≥ 5 admit a Simson line, and if so, under what conditions?
• RQ2 What geometric properties do equidistant Simson polygons share with parabolas, and how do they approximate the parabola?
• RQ3 How do discrete analogs of the optical property, Archimedes’ Lemma, and Lambert’s Theorem emerge in equidistant Simson polygons?
• RQ4 What is the role of the Miquel point in characterizing Simson polygons of higher-order?
• RQ5 How does the limit of equidistant Simson polygons recover classical parabolic theorems?

Key findings

• No convex n-gon with n ≥ 5 admits a Simson point, as proven by induction and contradiction using Miquel point and cyclic quadrilateral properties.
• Equidistant Simson polygons are constructed for every n ≥ 3, with their midpoints lying on a parabola y = x²/(4s) and their side slopes matching the parabola’s derivative at those points.
• The reflection of a line orthogonal to the Simson line at the midpoint of each side passes through the focus S, establishing a discrete analog of the optical property of parabolas.
• The vertices opposite the bases of Archimedes triangles formed by non-adjacent sides of equidistant Simson polygons lie on a line orthogonal to the Simson line, mirroring the classical result.
• The circumcircle of any triangle formed by three sides of a Simson polygon passes through the Simson point, generalizing the Simson-Wallace Theorem.
• Taking the limit of equidistant Simson polygons recovers Lambert’s Theorem: the focus of a parabola lies on the circumcircle of any triangle formed by three tangents.