Nine-Point Circle — Definition, Formula & Examples

The nine-point circle is a circle that passes through nine specific points associated with any triangle: 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.

Given a triangle $ABC$ with orthocenter $H$ , the nine-point circle is the unique circle passing through the midpoints of sides $\overline{AB}$ , $\overline{BC}$ , and $\overline{CA}$ ; the feet of the altitudes from $A$ , $B$ , and $C$ ; and the midpoints of $\overline{AH}$ , $\overline{BH}$ , and $\overline{CH}$ . Its radius equals half the circumradius of the triangle.

Key Formula

r_9 = \frac{R}{2}

Where:

$r_9$ = Radius of the nine-point circle
$R$ = Circumradius (radius of the circumscribed circle) of the triangle

How It Works

To find the nine-point circle, identify the nine special points on any triangle. The center of the nine-point circle, often labeled

N

, is the midpoint of the segment joining the orthocenter

H

and the circumcenter

O

. The radius of the nine-point circle is exactly

\frac{R}{2}

, where

R

is the circumradius. This relationship holds for every triangle, whether acute, right, or obtuse.

Worked Example

Problem: A triangle has a circumradius of R = 10. Find the radius of its nine-point circle and determine how many special points lie on it.

Step 1: Apply the nine-point circle radius formula.

r_9 = \frac{R}{2} = \frac{10}{2} = 5

Step 2: Count the nine points: 3 side midpoints + 3 altitude feet + 3 midpoints of vertex-to-orthocenter segments.

3 + 3 + 3 = 9

Answer: The nine-point circle has radius 5 and passes through exactly 9 special points of the triangle.

Why It Matters

The nine-point circle appears in competition mathematics and proof-based geometry courses. It connects several triangle centers — the orthocenter, circumcenter, and centroid — through the Euler line, making it a cornerstone result in advanced Euclidean geometry.

Common Mistakes

Mistake: Confusing the nine-point circle with the circumscribed circle (circumcircle).

Correction: The circumcircle passes through the three vertices of the triangle and has radius

R

. The nine-point circle passes through nine different points and has radius

\frac{R}{2}

. They are distinct circles with different centers.