Pompeiu-like theorems for the medians of a simplex. (English) Zbl 1392.51004
One theorem of Pompeiu states that the distances from the vertices of a regular triangle to an arbitrary point form a triangle. Another Pompeiu theorem states that the medians also form a triangle. The authors of this paper describe two generalizations of Pompeiu’s theorem for a regular \(n\)-simplex and explore the possibility of obtaining similar generalisations for the medians and for the generalized medians. They define the Fiedler property and Gerber property and discuss the relation between these properties. At the end of the article, the authors pose the question if the Pompeiu-like theorems hold for the other cevians. For example, the altitudes or the angle bisectors of a triangle form a triangle?
Reviewer: Sándor Nagydobai Kiss (Satu Mare)
MSC:
| 51M04 | Elementary problems in Euclidean geometries |
| 51M16 | Inequalities and extremum problems in real or complex geometry |
| 51M15 | Geometric constructions in real or complex geometry |
| 51M25 | Length, area and volume in real or complex geometry |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
Keywords:
content; facet; generalized medians; simplex; isodynamic simplex; medians; Pompeiu’s theoremReferences:
| [1] | G H. A L-A FIFI, M. H AJJA,AND A. H AMDAN, {\it Another n -dimensional generalization of Pompeiu’s} {\it theorem}, Amer. Math. Monthly, to appear. |
| [2] | P. S. B ULLEN, {\it A Dictionary of Inequalities}, second edition, Taylor & Francis Group, LLC, Boca Raton, Florida, 2015. |
| [3] | M. F IEDLER, {\it Isodynamic systems in Euclidean spaces and an n -dimensional analogue of a theorem} {\it by Pompeiu}, ˇCasopis pro pˇestov´an´ı matematiky 102 (1977), 370-381. · Zbl 0383.51014 |
| [4] | L. G ERBER, {\it The orthocentric simplex as an extreme simplex}, Pacific J. Math. 56 (1975), 97-111. · Zbl 0303.52004 |
| [5] | M. H AJJA, {\it On nested sequences of triangles}, Results Math. 54 (2009), 289-299. · Zbl 1183.51006 |
| [6] | M. H AJJA, {\it The sequence of generalized median triangles and a new shape function}, J. Geom. 96 (2009), 71-79. · Zbl 1204.51020 |
| [7] | M. H AJJA AND M. H AYAJNEH, {\it Impurity of the corner angles in certain special families of simplices}, J. Geom. 105 (2014), 539-560. · Zbl 1309.51012 |
| [8] | M. H AJJA, M. H AYAJNEH,AND H. M ARTINI, {\it More characterizations of certain special families of} {\it simplices}, Results Math. 69 (2016), 23-47. · Zbl 1335.52024 |
| [9] | G. H EINDL, {\it How to compute a triangle with prescribed lengths of its internal angle bisectors}, Forum Geom. 16 (2016), 407-414. · Zbl 1362.51007 |
| [10] | R. H ONSBERGER, {\it Mathematical Delights}, The Dolciani Mathematical Expositions, no. 28, MAA, Washington, D. C., 2004. · Zbl 1061.00003 |
| [11] | S. I ZUMI, {\it Sufficiency of simplex inequalities}, Proc. Am. Math. Soc. 144 2016, 1299-1307. · Zbl 1336.51014 |
| [12] | R. A. J OHNSON, {\it Advanced Euclidean Geometry}, Dover Publications, Inc., New York, 1929. · Zbl 0090.37301 |
| [13] | P. M IRONESCU AND L. P ANAITOPOL, {\it The existence of a triangle with prescribed angle bisector} {\it lengths}, Amer. Math. Monthly 101 (1994), 58-60. · Zbl 0802.51017 |
| [14] | S. F. O SINKIN, {\it On the existence of a triangle with prescribed angle bisector lengths}, Forum Geom. 16 (2016), 399-405. · Zbl 1362.51017 |
| [15] | S. S AVCHEV AND T. A NDREESCU, {\it Mathematical Miniatures}, Anneli Lax New Mathematical Library, No. 43, MAA, Washington, D. C., 2003. · Zbl 1012.00004 |
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.
