  In this paper we will do the following: (1) show how to geometrically define
multiplication, using only basic plane geometry, independently of area and any
notion of similar triangles; (2) prove all the properties of multiplication
using only the axioms of plane geometry and the geometric definition of
multiplication; (3) explain how the geometric definition of multiplication
relates to the area of a right triangle (or rectangle); and (4) explain how by
using only the geometric definition of multiplication and the Pythagorean
Theorem one can prove that two triangles have the same angles if and only if
the lengths of their corresponding sides are proportional. The interesting and
surprising thing, from a pedagogical and/or mathematical point of view, is that
all of these results can be proven using only simple geometry (no limits
needed). As we shall see, parallel lines in our geometric approach will play a
role similar to limits in the standard algebraic approach. We believe that both
prospective K-12 mathematics teachers and STEM students could especially
benefit by being exposed to the material in this paper before graduating from
university.
