### Abstract

On the basis of the theory ε^{-} of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: "In any right triangle the hypotenuse is greater than the leg" and "If ∠AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD." This represents an attempt to split the Pasch axiom with respect to ε^{-}. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.

Original language | English (US) |
---|---|

Pages (from-to) | 126-130 |

Number of pages | 5 |

Journal | Journal of Geometry |

Volume | 56 |

Issue number | 1-2 |

State | Published - 1996 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**Splitting the pasch axiom.** / Pambuccian, Victor.

Research output: Contribution to journal › Article

*Journal of Geometry*, vol. 56, no. 1-2, pp. 126-130.

}

TY - JOUR

T1 - Splitting the pasch axiom

AU - Pambuccian, Victor

PY - 1996

Y1 - 1996

N2 - On the basis of the theory ε- of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: "In any right triangle the hypotenuse is greater than the leg" and "If ∠AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD." This represents an attempt to split the Pasch axiom with respect to ε-. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.

AB - On the basis of the theory ε- of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: "In any right triangle the hypotenuse is greater than the leg" and "If ∠AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD." This represents an attempt to split the Pasch axiom with respect to ε-. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.

UR - http://www.scopus.com/inward/record.url?scp=53349173792&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=53349173792&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:53349173792

VL - 56

SP - 126

EP - 130

JO - Journal of Geometry

JF - Journal of Geometry

SN - 0047-2468

IS - 1-2

ER -