The Euler line is a classical geometric object defined for non-equilateral triangles as the line passing through the centroid, circumcenter, and orthocenter. In this paper, we proposed a generalization of the Euler line to convex polygons. The approach was based on constructing families of triangles associated with a given polygon and defining averaged centers, namely, the average circumcenter, centroid, and orthocenter. We proved that, for convex quadrilaterals, these averaged points are collinear, thus naturally defining a Euler line for quadrilaterals. Several geometric properties were established, including characterizations for isosceles trapezoids, right trapezoids, and cyclic quadrilaterals. Furthermore, we extended the construction to convex polygons with an arbitrary number of sides and provide an algorithmic method to compute the generalized Euler line. These results extend classical triangle geometry to polygonal settings and provided a new framework for studying geometric structures through averaged centers.
Citation: Ricardo Velezmoro León, Carlos Arellano Ramírez, Robert Ipanaqué Chero, Vanessa Silupu Ortega, Hebert Espino Aguirre, Eder Escobar Gómez. On a generalization of the Euler line for convex polygons via averaged triangle centers[J]. AIMS Mathematics, 2026, 11(7): 20291-20308. doi: 10.3934/math.2026824
The Euler line is a classical geometric object defined for non-equilateral triangles as the line passing through the centroid, circumcenter, and orthocenter. In this paper, we proposed a generalization of the Euler line to convex polygons. The approach was based on constructing families of triangles associated with a given polygon and defining averaged centers, namely, the average circumcenter, centroid, and orthocenter. We proved that, for convex quadrilaterals, these averaged points are collinear, thus naturally defining a Euler line for quadrilaterals. Several geometric properties were established, including characterizations for isosceles trapezoids, right trapezoids, and cyclic quadrilaterals. Furthermore, we extended the construction to convex polygons with an arbitrary number of sides and provide an algorithmic method to compute the generalized Euler line. These results extend classical triangle geometry to polygonal settings and provided a new framework for studying geometric structures through averaged centers.
| [1] | C. Kimberling, Triangle centers and central triangles, Congressus Numerantium, 129 (1998), 1–295. |
| [2] | R. A. Johnson, Advanced Euclidean Geometry, Dover Publications, New York, 2007. |
| [3] | H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, Mathematical Association of America, Washington, DC, 1967. https://doi.org/10.5948/UPO9780883859346 |
| [4] | E. W. Weisstein, Euler line, From MathWorld–A Wolfram Web Resource. Available from: https://mathworld.wolfram.com/EulerLine.html |
| [5] |
D. Kalman, An elementary proof of Euler's theorem on the Euler line, Mathematics Magazine, 70 (1997), 97–101. https://doi.org/10.1093/forestry/70.1.97 doi: 10.1093/forestry/70.1.97
|
| [6] | M. Dalcín, Euler line in quadrilaterals, Matemática, Educación e Internet, 7 (2006), 1–14. |
| [7] |
M. de Villiers, The Euler line and nine-point circle for quadrilaterals, Mathematical Gazette, 86 (2002), 38–44. https://doi.org/10.1006/gyno.2002.6713 doi: 10.1006/gyno.2002.6713
|
| [8] | R. Velezmoro, R. Ipanaqué, M. Fernández, J. Jiménez, Construction of polyhedra whose vertices lie on a curve on a lemniscatic torus, in International Conference on Computational Science and Its Applications, Springer, Cham, 2021, 3–17. https://doi.org/10.1007/978-3-030-86979-3 |
| [9] | Ch. Lehmann, Geometría analítica, Limusa, México, 1989. |
| [10] | C. Bradley, The algebra of geometry: Cartesian, areal and projective coordinates, Highperception, 2007. |
| [11] | L. Cordero, M. Fernández and A. Gray, Geometría diferencial de curvas y superficies con Mathematica, Addison-Wesley, 1995. |
| [12] | R. Figueroa, Vectores y matrices, Editorial América, Lima, 1993. |
| [13] | A. Venero, Introducción al análisis matemático, Lima, 1995. |
| [14] | Wolfram Research, Inc., Mathematica, Version 13.0, Champaign, IL, 2022. |
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Ricardo Velezmoro León, Carlos Arellano Ramírez, Robert Ipanaqué Chero, Vanessa Silupu Ortega, Hebert Espino Aguirre, Eder Escobar Gómez. On a generalization of the Euler line for convex polygons via averaged triangle centers[J]. AIMS Mathematics, 2026, 11(7): 20291-20308. doi: 10.3934/math.2026824
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