The Nine-Point Circle

An animated demonstration showing that nine special points of a triangle — the three side midpoints, three altitude feet, and three orthocenter midpoints — all lie on a single circle called the nine-point circle.

Keywords: nine-point circle, Feuerbach circle, triangle centers, midpoints, altitudes, orthocenter

Prerequisites: orthocenter-proof, circumcenter-proof · Difficulty: intermediate

For any triangle, nine special points all lie on a single circle, called the nine-point circle, whose radius is half the circumradius. The nine points fall into three groups of three: the side midpoints, the altitude feet, and the midpoints of the segments from each vertex to the orthocenter.

Strategy: mark the three groups of points in turn, locate the nine-point center $N$ as the midpoint of segment $O H$ on the Euler line, then draw the circle of radius $R /2$ at $N$ and verify all nine points lie on it.

Side Midpoints

We begin with triangle $A B C$ and mark the midpoints of each side:

$M_{A}$ : midpoint of $B C$
$M_{B}$ : midpoint of $C A$
$M_{C}$ : midpoint of $A B$

These form the medial triangle. These are the first three of our nine special points.

Altitude Feet

Next, we find the orthocenter $H$ where the three altitudes meet, and mark the altitude feet — the points where each altitude meets the opposite side:

$F_{A}$ : foot of altitude from $A$
$F_{B}$ : foot of altitude from $B$
$F_{C}$ : foot of altitude from $C$

These are the second three of our nine special points.

Orthocenter Midpoints

Finally, we mark the midpoints of segments from each vertex to the orthocenter:

$E_{A}$ : midpoint of $A H$
$E_{B}$ : midpoint of $B H$
$E_{C}$ : midpoint of $C H$

These are the third three of our nine special points.

The Nine-Point Center

Construct the circumcenter $O$ and draw the segment from $O$ to the orthocenter $H$ . This segment (and the line it lies on) is called the Euler line.

Let $N$ be the midpoint of $O H$ . It can be shown that all nine points lie on the circle of radius $R /2$ centered at $N$ (where $R$ is the circumradius); for this reason $N$ is called the nine-point center.

The Nine-Point Circle

It can be shown that all nine special points lie on the circle of radius $R /2$ centered at $N$ :

the three side midpoints,
the three altitude feet, and
the three orthocenter midpoints.

This circle is called the nine-point circle.

All nine special points lie on the nine-point circle:

Center: $N$ , the midpoint of segment $O H$ (the Euler line).
Radius: $R /2$ , where $R$ is the circumradius.

This construction works for any triangle.

The nine-point circle is also tangent to the incircle and all three excircles — a result known as Feuerbach's theorem, covered in a later scene.

Notes

The Nine Points

Group 1: Side midpoints (3 points)

$M_{A}$ : midpoint of $B C$
$M_{B}$ : midpoint of $C A$
$M_{C}$ : midpoint of $A B$

Group 2: Altitude feet (3 points)

$F_{A}$ : foot of altitude from $A$ to $B C$
$F_{B}$ : foot of altitude from $B$ to $C A$
$F_{C}$ : foot of altitude from $C$ to $A B$

Group 3: Orthocenter midpoints (3 points)

$E_{A}$ : midpoint of segment $A H$
$E_{B}$ : midpoint of segment $B H$
$E_{C}$ : midpoint of segment $C H$

(where $H$ is the orthocenter)