On a generalization of the Euler line for convex polygons via averaged triangle centers

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.