On the Thales Theorem in the Iso-Taxicab Plane

Abstract

There are non-Euclidean geometries that are both accessible in a very concrete form and close to Euclidean geometry in their basic structure. In this paper, we study the Iso-taxicab analogues of the intercept theorem, also known as Thales's theorem is an important theorem in elementary geometry.

Keywords

• The plane $\mathbb{R}_{\pi 3}^{2}$
• Thales theorem
• non-Euclidean geometry

References

1. Akça, Z., Bayar, A., & Ekmekçi, S. (2007). The norm in CC-plane geometry. Pi Mu Epsilon Journal, 12(6), 321-324.
2. Bayar, A., Ekmekçi, S., & Z. Akça (2008). On the plane geometry with generalized absolute value metric. Mathematical problems in Engineering, 1-8.
3. Bayar, A., Ekmekçi, S., & Özcan, M. (2009). On trigonometric functions and cosine and sine rules in taxicab plane. International Electronic Journal of Geometry, 2(1), 17-24.
4. Bayar, A., Ekmekçi, S., & Öztürk, ̇I. (2015). On complex numbers and Taxicab plane. Mathematical Sciences and Applications E-Notes, 3(1), 58-64.
5. Çolakoğlu, H.B., & Kaya, R. (2008). Taxicab versions of the Pythagorean theorem. Pi Mu Epsilon Journal, 12(9), 535-539.
6. Ekmekçi, S., Akça, Z., & Altıntaş K. (2015). On trigonometric functions and norm in the generalized taxicab metric. Mathematical Sciences And Applications E-Notes, 3(2), 27-33.
7. Özcan, M., Ekmekçi, S., & Bayar, A. (2002). A note on the variation of the taxicab lengths under rotations. Pi Mu Epsilon Journal, 11(7), 381-384.
8. Sowell, K. O. (1989). Taxicab geometry—a new slant. Mathematics Magazine, 62(4), 238-248.

9. Kocayusufoğlu, İ. (2000). Trigonometry on Iso-taxicab geometry. Mathematical and Computational Applications, 5(3), 201-212.
10. Bayar, A., & Kaya, R. (2011). On isometries of $\mathbb{R}_{\pi n}^{2}.$. Hacettepe Journal of Mathematics and Statistics, 40(5), 673-679.
11. Akça, Z., & Nazlı, S. (2022). On the versions in the plane $\mathbb{R}_{\pi3}^{2}$ of some Euclidean theorems. New Trends in Mathematical Sciences, 10(1), 20-27.
12. Nazlı, S., & Akça, Z. (2020). On the trigonometric functions in $\mathbb{R}_{\pi 3}^{2}$. Konuralp Journal of Mathematics, 8(2), 429-437.
13. Akça, Z., & Nazlı, S. (2022). On the shortest distance of a point to the line in the plane $\mathbb{R}_{\pi 3}^{2}$. New Trends in Mathematical Sciences, 10(4), 128-132.