Nine-Point Center (N)

The nine-point center N is the center of the circle through the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from each vertex to the orthocenter. N is also the midpoint of segment OH on the Euler line, which additionally contains the centroid G.

Keywords: nine-point center, nine-point circle, Euler line, midpoints, centroid, incenter, triangle centers

Prerequisites: circumcenter, orthocenter · Difficulty: intermediate

This is one of the most amazing features of a triangle! There is a special circle, called the nine-point circle, that passes through nine important points on the triangle. The center of this circle is the nine-point center.

What are the nine points?

The nine points are:

  1. The three midpoints of the sides. (The halfway points of each side of the triangle).
  2. The three "feet" of the altitudes. (An altitude is a line from a vertex that is perpendicular to the opposite side. The "foot" is the point where the altitude touches that side).
  3. The three midpoints of the segments connecting each vertex to the orthocenter. (The orthocenter is where the three altitudes intersect).

It's incredible that one circle can go through all nine of these points for any triangle!

The Nine-Point Center and the Euler Line

The nine-point center $N$ is the center of this amazing circle. It also has a special location: it's the midpoint of the segment connecting the orthocenter $H$ and the circumcenter $O$. That segment is the Euler line, and the centroid $G$ lies on it too, with $OG:GH = 1:2$ and $ON:NH = 1:1$.

The incenter $I$ (center of the incircle) is usually not on the Euler line — but the incircle is tangent to the nine-point circle (Feuerbach's theorem).