Nine-Point Circle

One circle through nine triangle points, centered on the Euler line.

The idea

Theorem (Nine-point circle theorem).

In any triangle, nine particular points lie on one circle: the three midpoints of the sides; the three feet of the altitudes, where each altitude meets the line of the opposite side; and the three Euler points, the midpoints of the segments joining each vertex to the orthocenter $H$, the common point of the three altitudes.

The circle is the nine-point circle; write $N$ for its center and $N_r$ for its radius. The theorem assumes nothing about the triangle. In a scalene triangle the nine points are nine distinct points, produced by three unrelated constructions, and one circle passes through all of them.

The circle's size and position come from the circumcircle, the circle through the three vertices, with center $O$ and radius $R$.

Proposition.

The nine-point circle has radius $N_r = \tfrac{1}{2} R$, and its center $N$ is the midpoint of the segment $OH$.

In particular $N$ lies on the Euler line, the line through $O$, the centroid, and $H$.

Ways to work on it

• Walkthrough. Locate the nine special points that share one circle, then find its radius and center.
• Practice. Find the nine-point radius from the circumradius.
• Hardest. Locate the nine-point center on the Euler line and test a point.

Not sure where to start? Take the ten-question placement test.