**John Donnelly,**

A Model of Nowhere Geodesic Plane Geometry in which the Triangle Inequality
Fails Everywhere,

Forum Geometricorum, 18 (2018) 275-296.

Abstract. Continuing with the results
from an earlier paper, we construct a model **M_2** of plane geometry that satisfies all of Hilbert's axioms for
the euclidean plane (with the
exception of Sided-Angle-Side), yet in which the geodesic line segment
connecting any two points *A *and* B* is never the shortest path from *A *to*
B*. Moreover, in the model **M_2**, the triangle inequality always
fails for any triple of noncollinear points.

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Geom., 18 (2018) Table of Contents