What is the shape of a triangle?

Abstract

We consider the problem of describing the shape of a triangle $'Delta ABC$ by a single complex number $'sigma( 'Delata ABC)$ , which we call the shape invariant of the triangle.After giving a simple algebraic definition of $'sigma$, we prove a surprising geometric description: modulo a symmetry group of order 6, $'sigma$ is the location of the orthocenter of the triangle, atter it is rescaled so that the vertices lie on the unit circle and rotated so that an altitude of its Morley triangle points in the direction of the positive x-axis. We find the set of all possible values of $'sigma$, and discuss how the value of $'sigma$ determines the scaleneness and acuteness of $Delta ABC$. Finally, we give formulas for the scaleneness and acuteness in terms of the side lengths or angles of $'Delta ABC$, and compute some numerical examples where the angles are unusual rational multiples of $'pi$.