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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