Euler line

The line containing a triangle’s circumcenter, centroid, and orthocenter. In a non-equilateral triangle, these centers are collinear, with the centroid two-thirds of the way from the orthocenter to the circumcenter.

Circumcenter: The center of the circle passing through a triangle’s three vertices. It is one endpoint of the segment whose line also contains the centroid and orthocenter.
Triangle median: A segment joining a triangle vertex to the midpoint of the opposite side. The medians intersect at the centroid, one of the Euler line’s defining centers.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped analysis, number theory, mechanics, and geometry. He established the line’s place in triangle geometry and gave it its name.
Orthic triangle: The triangle formed by the feet of a triangle’s three altitudes. Its vertices are tied to the altitudes that define the Euler line’s orthocenter.
Centroid: The intersection of a triangle’s three medians, which divides each median in a two-to-one ratio. It lies on the Euler line two-thirds of the way from the orthocenter to the circumcenter.
Triangle altitude: A perpendicular from a triangle vertex to the line containing the opposite side. The altitudes intersect at the orthocenter, another defining center.
William Chapple: An English clergyman and mathematician who published an early proof of the relation between a triangle’s incenter and circumcenter. His 1746 result preceded Euler’s work on the line joining those centers.
Medial triangle: The triangle formed by joining the midpoints of a triangle’s three sides. Its centroid and center relations provide a scaled setting for studying the original triangle’s centers.
Orthocenter: The point where a triangle’s three altitudes intersect. It is the other defining endpoint of the Euler line’s center alignment.
Circumcircle: The unique circle passing through all three vertices of a nondegenerate triangle. Its center is the circumcenter, which fixes one of the Euler line’s principal points.

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