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1. INTRODUCTION. Discussions that led to this paper began during an electronic version of the Secondary School Teachers Program, a part of the Park City Mathematics Institute (PCMI), offered at the University of Cincinnati in the summer of 2006 for local high school mathematics teachers. The authors were working with the teachers on daily problem sets provided by PCMI; complex numbers and polynomials were a theme throughout the problems. The second author raised the question of geometric connections between the triangle whose vertices are the roots of a cubic polynomial and the roots of the derivative of the polynomial. He had noticed some relationships, such as that the roots of the derivative were inside the triangle, by performing experiments using Geometers Sketchpad. The second author was able to justify some of the observed relations. We continued to investigate the matter for the three weeks …