A comprehensive analysis of the axiomatic frameworks in which Fagnano’s theorem holds

Proofs of Fagnano’s Theorem (FT)—which is known to hold in plane absolute geometry, a fact emphasized by P. Szász and implicit in L. Fejér’s proof—under various axiomatic assumptions constitute the subject of this paper. It starts with two reflection-geometric proofs, valid in ordered metric (Bachmann) planes with free mobility, modeled on the proofs by P. Szász and L. Fejér, followed by a proof in standard ordered Hjelmslev groups with free mobility. Two minimalist axiom systems—one for non-elliptic metric planes with some rudimentary order axioms, the other for non-elliptic metric planes in which there are motions that move lines that intersect into each other, together with more order axioms—are the result of a search for minimal assumptions ensuring the validity of FT. FT is shown to be false in Galilean geometry and, appropriately restated, holds in elliptic ordered geometry.