On the existence of a equilateral triangle in \(H\)-planes (Q1281742)

This interesting paper contains a proof for the existence of an equilateral triangle in absolute planes (satisfying Hilbert's axioms I 1-3, II, III). In particular, it is shown how to construct such a triangle with ruler and gauge. However, it is in general not possible to find an equilateral triangle with a given base, even if the plane is satisfying Bachmann's ``Lotschnittaxiom'' or the Archimedian axiom. The non--existence is shown for an absolute plane over the Pythagorean hull of \({\mathbb Q}(t)\) (with the usual ordering) and an absolute plane over a super-Pythagorean field (allowing exactly two orderings), respectively.