Stewart’s Theorem and Median Property in the Galilean Plane

Year 2023, Volume: 9 Issue: 2, 276 - 282, 30.06.2023

Abdulaziz Açıkgöz , Nilgün Sönmez

https://doi.org/10.28979/jarnas.1190619

Abstract

Galilean plane can be introduced in the affine plane, as in Euclidean plane. This means that the concepts of lines, parallel lines, ratios of collinear segments, and areas of figures are significant not only in Euclidean plane but also in Galilean plane. The Galilean plane 𝐺2 is almost the same as the Euclidean plane. The coordinates of a vector 𝑎 and the coordinates of a point 𝐴 (defined as the coordinates of 𝑂𝐴, where 𝑂 is the fixed origin) are introduced in Galilean plane in the same way as in Euclidean geometry. The galilean lines are the same. All we need add is that we single out special lines with special direction vectors in Galilean plane. We should attention that these two types of galilean lines cannot be compared. The difference between Euclidean plane and Galilean plane is the distance function. Thus, we can compare the many theorems and properties which is included the concept of distance in these geometries. The theorems and the properties of triangles in the Euclidean plane can be studied in the Galilean plane. Therefore, in this study, we give the Galilean-analogues of Stewart’s theorem and median property for the triangles whose sides are on ordinary lines.

Keywords

Galilean distance, Galilean plane, Galilean triangles, Median property, Stewart's theorem

References

• Yaglom, I.M. (1979). A Simple Non-Euclidean Geometry and Its Physical Basis. New York, Springer-Verlag.
• Ozcan, M., & Kaya R. (2003). Area of a Triangle in Terms of the Taxicab Distance. Missouri J. Of Math. Sci., vol. 15, pp. 178–185.
• Kurudirek A., & Akca, H. (2015). On the Concept of Circle and Angle in Galilean Plane. Open Access Library Journal, 2: e1256. http://dx.doi.org/10.4236/oalib.1101256.
• Akar, M., Yuce S., & Kuruoglu, N. (2013). One-Parameter Planar Motion on the Galilean Plane. International Elektronic Journal of Geometry, Volume 6, No: 1, pp. 79-88.
• Kaya, R., & Colakoglu, H.B. (2006). Taxicab Version of Some Euclidean Theorem. Int. Jour. of Pure and Appl. Math. (IJPAM) 26, 1, 69-81.
• Gelisgen, O., & Kaya, R. (2013). The Alpha-Version of the Stewart’s Theorem. Demonstratıo Mathematıca, Volume: XLVI, No: 4, pp. 795-808. https://doi.org/10.1515/dema-2013-048.
• Gelisgen, O., & Kaya, R. (2009). The CC-version of the Stewart’s Theorem. Balkan Society of Geometries Geometry Balkan Press, Volume: 11, pp. 68-77.