The Euler line of a triangle passes through a triangle's orthocenter, centroid and circumcenter. The orthocenter is the intersection of the three altitudes of the triangle. The centroid is the intersection of the three medians (the lines connecting each vertex to the midpoint of the opposite side). The circumcenter is the center of the circumscribed triangle.
Let be a triangle in the plane. This Demonstration shows how the angle of the Euler line of changes as the vertex rotates about the midpoint of on a circle of variable radius . An accompanying graph plots versus the rotation of along its path (measured in radians). The relationship between the two angles changes with .
Three ranges for the circle radius produce qualitatively different results. For and , the Euler line "wobbles" and never completes a rotation. For , the Euler line completes two full rotations as goes around the circle once. At and , the Euler line completes one full rotation. You can see this when the jumps on the graph (which indicate a 180° rotation) converge to two jumps instead of the four in the middle range.
Torrey Pines High School, Advanced Topics in Mathematics II, 2016–2017