Theorems Regarding Points on the Euler Line

The Euler line of a triangle passes through several important points, including three specific triangle centers: the centroid, orthocenter, and circumcenter. Each of these centers is the intersection of lines related to the triangle, mainly its medians, altitudes, and perpendicular bisectors, respectively. The theorems presented here involve creating a triangle and picking a specific point. Then after connecting the triangle’s vertices to that point, thereby creating additional triangles, I establish connections to either the centroids, orthocenters, or circumcenters of the new triangles