  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.
