Topic
Pons asinorum
About: Pons asinorum is a research topic. Over the lifetime, 7 publications have been published within this topic receiving 32 citations. The topic is also known as: pons asinorum.
Papers
TL;DR: In this article, it was shown that a reasonably satisfactory generalization holds for all, not necessarily orthocentric, tetrahedra, and that a stronger statement than that given in [14] holds for orthogonality tetrahedral tetras.
Abstract: Proposition 5 of Book I of Euclid’s Elements, better known as the Pons Asinorum or the Asses’ Bridge, and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. A generalization of this statement to higher dimensional d-simplices is considered in [14], where it is shown that such a generalization holds only if the underlying d-simplex is orthocentric. In this paper, it is shown that a reasonably satisfactory generalization holds for all, not necessarily orthocentric, tetrahedra. It is also shown that a stronger statement than that given in [14] holds for orthocentric tetrahedra.
10 citations
TL;DR: The Pons Asinorum, or the Bridge of Asses, refers to Proposition 5 of Book I of Euclid's Elements and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal as mentioned in this paper.
Abstract: The Pons Asinorum , or the Bridge of Asses , refers to Proposition 5 of Book I of Euclid’s Elements . This proposition and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. Analogues of these propositions for higher dimensional d -simplices are considered in this paper, and satisfactory results are obtained for orthocentric d -simplices. These results do not hold for non-orthocentric d -simplices, thus supporting the point of view that orthocentric d -simplices and not arbitrary ones are the adequate generalization of triangles.
9 citations
TL;DR: Abu-Saymeh and Hayajneh as discussed by the authors showed that if an angle in a triangle is increased without changing the lengths of its arms, then the length of the opposite side increases, and conversely.
8 citations
1 Oct 2012
TL;DR: A satisfactory generalization of this statement to general tetrahedra and a strong generalization to orthocentric d-simplices, d ≥ 3, were obtained in this paper.
Abstract: Proposition 5 of Book I of Euclid’s Elements, better known as the Pons Asinorum or the Bridge of Asses, and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. A satisfactory generalization of this statement to general tetrahedra and a strong generalization to orthocentric tetrahedra are obtained in Hajja (J Geom 93:71–82, 2009a), and generalizations to orthocentric d-simplices, d ≥ 3, are obtained in Hajja (Stud Sci Math Hungarica 46:263–273, 2009b). In this paper, the strong generalization is seen to hold for a fairly large family of tetrahedra that are referred to as distinguished. These include acute and rectangular orthocentric tetrahedra as well as circumscriptible and isogonic tetrahedra. It is also seen that the strong generalization fails for isodynamic tetrahedra and that even a weak generalization fails for simplices in dimension four.
4 citations
2 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2017 | 1 |
| 2015 | 1 |
| 2012 | 2 |
| 2009 | 2 |
| 1988 | 1 |