Nine-point center
The nine-point center is a special center of any triangle. It is the center of the nine-point circle, which passes through nine important points: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from each vertex to the orthocenter.
Where is it? The nine-point center, usually denoted N, lies on the Euler line—the line through the triangle’s orthocenter H and circumcenter O. In fact, N is the midpoint of OH. The centroid G also lies on this line, two-thirds of the way from H to O. Because H, O, G, and N are collinear, knowing any two of these centers lets you determine the others.
What does N do? It is the circumcenter of three derived triangles: the medial triangle (formed by the midpoints of the sides), the orthic triangle (the feet of the altitudes), and the Euler triangle (the midpoints of AH, BH, CH). More generally, N is the circumcenter of any triangle made from three of the nine defining points on the nine-point circle.
Other facts: The four points A, B, C, H form an orthocentric system, and the Euler lines of the four triangles ABC, ABH, BCH, and CAH all meet at N. The nine-point circle is connected to reflections of the triangle’s midpoints, and Lester’s theorem says N lies on a circle with the circumcenter and the two Fermat points. The Kosnita point is the isogonal conjugate of N. In obtuse triangles, one of the barycentric coordinates of N can be negative, which means N lies outside the triangle.
This page was last edited on 3 February 2026, at 19:38 (CET).