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The equivalence of Side-Angle-Side with Side-Side-Side and the general triangle inequality in the absolute plane

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Abstract

We show that if we start with an absolute plane, remove Side-Angle-Side as an axiom, and replace it with Side-Side-Side together with a weakened version of the triangle inequality as new axioms, then the new resulting axiom system is also an absolute plane. In particular, we show that Side-Angle-Side still holds in the new axiom system. In addition, we give a new proof of the two-circle theorem which does not depend on Side-Angle-Side, but instead uses Side-Side-Side and the weakened version of the triangle inequality.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern Indiana, 8600 University Boulevard, Evansville, IN, 47712, USA

    John Donnelly

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  1. John Donnelly
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Correspondence to John Donnelly.

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Donnelly, J. The equivalence of Side-Angle-Side with Side-Side-Side and the general triangle inequality in the absolute plane. J. Geom. 104, 265–275 (2013). https://doi.org/10.1007/s00022-013-0163-5

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  • DOI: https://doi.org/10.1007/s00022-013-0163-5

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