On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas

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, $n\ge 5$, admits a Simson Line. We then determine a property which characterizes Simson $n$-gons and show that one can be constructed for every $n\ge 3$. 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.