The Hajja–Martini inequality in a weak absolute geometry

Abstract

Solving a problem left open in Hajja and Martini (Mitt. Math. Ges. Hamburg 33:135–159, 2013), we prove, inside a weak plane absolute geometry, that, for every point P in the plane of a triangle ABC there exists a point Q inside or on the sides of ABC which satisfies: 1 $$\begin{aligned} AQ \le AP, BQ \le BP, CQ\le CP. \end{aligned}$$ If P lies outside of the triangle ABC, then Q can be chosen to both lie inside the triangle ABC and such that the inequalities in (1) are strict. We will also provide an algorithm to construct such a point Q.