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Manifold-based Proving Methods in Projective Geometry

Michael Martin Katzenberger, Jürgen Richter-Gebert

Abstract

This article compares different proving methods for projective incidence theorems. In particular, a technique using quadrilateral tilings recently introduced by Sergey Fomin and Pavlo Pylyavskyy is shown to be at most as strong as proofs using bi-quadratic final polynomials and thus, also proofs using Ceva-Menelaus-tilings. Furthermore, we demonstrate the equivalence between quadrilateral-tiling-proofs and proofs using exclusively Menelaus configurations. We exemplify the transition between the proofs in several examples in 2D and in 3D.

Manifold-based Proving Methods in Projective Geometry

Abstract

This article compares different proving methods for projective incidence theorems. In particular, a technique using quadrilateral tilings recently introduced by Sergey Fomin and Pavlo Pylyavskyy is shown to be at most as strong as proofs using bi-quadratic final polynomials and thus, also proofs using Ceva-Menelaus-tilings. Furthermore, we demonstrate the equivalence between quadrilateral-tiling-proofs and proofs using exclusively Menelaus configurations. We exemplify the transition between the proofs in several examples in 2D and in 3D.
Paper Structure (16 sections, 3 theorems, 22 equations, 23 figures)

This paper contains 16 sections, 3 theorems, 22 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Binomial Proofs and Ceva-Menelaus-Proofs
  3. Binomial Proofs
  4. Ceva-Menelaus-Proofs
  5. Quad proofs á la Fomin and Pylyavskyy
  6. Extracting a binomial proof from a quad-proof
  7. Relations between the Tiling-Based Methods
  8. Quad-proofs and Ceva-Menelaus-proofs
  9. Ceva vs Menelaus
  10. Summary of the Results
  11. Two illustrative examples
  12. Desargues' Theorem
  13. Pappus' Theorem
  14. Three three-dimensional examples
  15. The Sixteen Point Theorem
  16. ...and 1 more sections

Key Result

theorem thmcountertheorem

(3D Menelaus): Let $p_1,\ldots,p_4$ be four non coplanar points in three-space. Consider a plane in general position $H$ and the intersections $q_i:=H\cap \overline{p_{i},p_{i+1}}$ (indices mod 4). Then we have (with oriented distances):

Figures (23)

  • Figure 1: Menelaus's and Ceva's theorem and their corresponding equations.
  • Figure 2: A coherent quadrangle and the underlying incidence structure.
  • Figure 3: Splitting a line-vertex by inserting a new quadrangle.
  • Figure 4: Translating a quad-proof into a pure Menelaus-proof.
  • Figure 5: Proof of Ceva's theorem using Menelaus's theorem twice.
  • ...and 18 more figures

Theorems & Definitions (3)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem