Eulerlinjen

Eulerlinjen (the Euler line) is a fundamental line in Euclidean plane geometry associated with a nondegenerate triangle. For any such triangle, the circumcenter O (center of the circumscribed circle), the centroid G (intersection of the medians), and the orthocenter H (intersection of the altitudes) are collinear. The line also passes through N, the center of the nine-point circle, which is the circle through the midpoints of the sides and the feet of the altitudes.

A key property is that O, G and H lie on one line known as the Euler line, with G dividing OH in a 1:2 ratio (OG:GH = 1:2); equivalently GH = 2·OG. The nine-point center N is the midpoint of OH and hence also lies on the Euler line. A related point, L, the De Longchamps point, is the reflection of H across O and also lies on the line.

The Euler line exists for all nondegenerate triangles in the Euclidean plane; its position varies with the triangle’s shape. When the triangle is acute, O and H lie inside the triangle; when obtuse, H lies outside. In a right triangle, O is the midpoint of the hypotenuse and H is at the right-angle vertex, but the Euler line still passes through the same centers. The concept is named after Leonhard Euler and is invariant under similarity transformations.