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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年3月12日

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  • Victor Pambuccian,
  • Horst Struve,
  • Rolf Struve

摘要

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.