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, it can be shown that nine special points all lie on a single circle. This circle is called the nine-point circle, and its radius is half the circumradius. The nine points fall into three groups of three.

The Nine Points

Group 1: Side midpoints (3 points)

• $M_A$: midpoint of $BC$
• $M_B$: midpoint of $CA$
• $M_C$: midpoint of $AB$

Group 2: Altitude feet (3 points)

• $F_A$: foot of altitude from $A$ to $BC$
• $F_B$: foot of altitude from $B$ to $CA$
• $F_C$: foot of altitude from $C$ to $AB$

Group 3: Orthocenter midpoints (3 points)

• $E_A$: midpoint of segment $AH$
• $E_B$: midpoint of segment $BH$
• $E_C$: midpoint of segment $CH$

(where $H$ is the orthocenter)

Side Midpoints

We begin with triangle $ABC$ and mark the midpoints of each side:

• $M_A$: midpoint of $BC$
• $M_B$: midpoint of $CA$
• $M_C$: midpoint of $AB$

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 $AH$
• $E_B$: midpoint of $BH$
• $E_C$: midpoint of $CH$

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 $OH$. 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.

Conclusion

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

• Center: $N$, the midpoint of segment $OH$ (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.