A theorem on equiareal triangles with a fixed base

Abstract

The statement: "Given two fixed points, $A$ and $C$, the locus of the midpoints of $AB$ and $CB$, when $B$ varies such that the area of triangle $ABC$ is constant, consists of two lines symmetric with respect to $AC$" is shown to be provable in very weak geometries, that is, Bachmann's non-elliptic metric planes in which every pair of points has a midpoint.