×

The Hajja-Martini inequality in a weak absolute geometry. (English) Zbl 1417.51009

Summary: Solving a problem left open in [M. Hajja and H. Martini, Mitt. Math. Ges. Hamb. 33, 135–159 (2013; Zbl 1303.51010)], we prove, inside a weak plane absolute geometry, that, for every point \(P\) in the plane of a triangle \(ABC\) there exists a point \(Q\) inside or on the sides of \(ABC\) which satisfies: \[ AQ \le AP, \ BQ \le BP, \ CQ\le CP. \tag{1}\] If \(P\) lies outside of the triangle \(ABC\), then \(Q\) can be chosen to both lie inside the triangle \(ABC\) and such that the inequalities in (1) are strict. We will also provide an algorithm to construct such a point \(Q\).

MSC:

51F05 Absolute planes in metric geometry
51M16 Inequalities and extremum problems in real or complex geometry
51M04 Elementary problems in Euclidean geometries
51M05 Euclidean geometries (general) and generalizations

Citations:

Zbl 1303.51010
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hajja, M., Martini, H.: Proposition 21 of Book I of Euclid’s Elements: variants, generalizations, and open questions. Mitt. Math. Ges. Hamburg 33, 135-159 (2013) · Zbl 1303.51010
[2] Pambuccian, V.: Prolegomena to any theory of proof simplicity. Philos. Trans. R. Soc. A 377, 20180035 (2019) · Zbl 1441.03007 · doi:10.1098/rsta.2018.0035
[3] Hociotă, I., Pambuccian, V.: Acute triangulation of a triangle in a general setting revisited. J. Geom. 102, 81-84 (2011) · Zbl 1246.51010 · doi:10.1007/s00022-011-0097-8
[4] Pambuccian, V.: The axiomatics of ordered geometry I. Ordered incidence spaces. Expo. Math. 29, 24-66 (2011) · Zbl 1227.51010 · doi:10.1016/j.exmath.2010.09.004
[5] Greenberg, M.J.: Euclidean and Non-Euclidean Geometries, 4th edn. W. H. Freeman, San Francisco (2008) · Zbl 1127.51001
[6] Pambuccian, V., Struve, H., Struve, R.: The Steiner-Lehmus theorem and “triangles with congruent medians are isosceles” hold in weak geometries. Beitr. Algebra Geom. 57, 483-497 (2016) · Zbl 1367.51008 · doi:10.1007/s13366-015-0278-y
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.