Lippmann’s axiom and Lebesgue’s axiom are equivalent to the Lotschnittaxiom

Abstract

We prove that both Lippmann’s axiom of 1906, stating that for any circle there exists a triangle circumscribing it, and Lebesgue’s axiom of 1936, stating that for every quadrilateral there exists a triangle containing it, are equivalent, with respect to Hilbert’s plane absolute geometry, to Bachmann’s Lotschnittaxiom, which states that perpendiculars raised on the two legs of a right angle meet. We also show that, in the presence of the Circle Axiom, the statement “There is an angle such that the perpendiculars raised on its legs at equal distances from the vertex meet” is equivalent to the negation of Hilbert’s hyperbolic parallel postulate.