||Recursive Napoleon-Like Constructions Investigated with a Symbolic Geometry System
We investigate limit behavior for the recursive application of a variety of constructions generalized from that of Napoleon’s Theorem. Napoleon’s Theorem states that if we draw an isosceles triangle with 120 degree angle at its apex on each of the three sides of a triangle, then the triangle formed by joining those apexes is equilateral. We generalize this construction by allowing any set of similar triangles. In particular we show that if the similar triangles have 120 degree angles at their apex, then recursive application of this pseudo Napoleon’s construction will, in the limit, construct a triangle congruent to Napoleon’s. We investigate this class of problems with a symbolic geometry system and a Computer Algebra System.