A Characterization of General Linear Transformations by Use of Symmedian Points of Triangles

Abstract

In this study, based on a geometric approach, we obtained a new characteristic of general linear transformations, also known as similarities, by use of symmedian points of triangles in complex z-plane.

Keywords

• General linear transformations
• Möbius transformations
• symmedian point of triangle

A characterization of general linear transformations by use of symmedian points of triangles

Abstract

In this study, based on a geometric approach, we obtained a new characteristic of general linear transformations, also known as similarities, by use of symmedian points of triangles in complex z-plane.

Keywords

• General linear transformations
• Möbius transformations
• symmedian point of triangle

References

1. [1] Li, B., Wang, Y., “A new characterization for isometries by triangles”, New York J. Math. 15 (2009) : 423-429.
2. [2] Lester, J., “Euclidean plane point-transformations preserving unit area or unit perimeter”, Arch. Math 45 (1985) : 561-564.
3. [3] Lester, J., “Martin’s theorem for Euclidean n-space and a generalization to the perimeter case”, J.Geom. 27 (1986) : 29-35.
4. [4] Benz, W., “Mappings preserving area 1 of triangles”, J.Geom. 79 (2004) : 27-30.
5. [5] Demirel, O., On mappings that preserve Fermat-Torricelli points”, Math. Commun. 26 (2021) : 21-27.
6. [6] Demirel, O., Aslan, L., Topal, D., “A characteristic of similarities by use of Steinhaus’ problem on partition of triangles”, Mathematical Sciences and Applications E- Notes 8(1) (2020) : 117-122.
7. [7] Aley, R.J., “Some properties of the symmedian point”, Proceedings o the Indiana Academy of Science 10 (1900) : 85-88.
8. [8] Haruki, H., Rassias, T.M., “A new characteristic of Möbius transformations by use of Apollonius points of triangles”, J. Math. Anal. Appl. 197(1) (1996) : 14-22.

9. [9] Pennisi, L.L., Gordon, L.I., Lasher, S., “Elements of Complex Variables”, Holt, Rinehart-Winston, New York, 1963.
10. [10] Nevanlinna, R., Paatero, V., “Introduction to Complex Analysis”, Addison-Wesley, New York, 1964.
11. [11] Nehari, Z. “Conformal Mapping”, McGraw-Hill Book, New York, 1952.