Learn/Geometry/Nine-Point Circle
Geometry • Topic 24

Nine-Point Circle

The Nine-Point Circle is one of the gems of triangle geometry. It passes through nine significant points defined by the triangle.

The Nine Points

Theorem (Nine-Point Circle)
For any triangle, the following nine points lie on a single circle:
  1. The midpoints of the three sides ().
  2. The feet of the three altitudes ().
  3. The midpoints of the segments from the orthocenter to the vertices ().

Properties

1. Center ()

The center of the Nine-Point Circle () lies on the Euler Line, exactly halfway between the Orthocenter () and the Circumcenter ().

2. Radius

The radius of the Nine-Point Circle is exactly half the circumradius ().

3. Feuerbach's Theorem

The Nine-Point Circle is tangent to the Incircle and the three Excircles.

Practice Problems

Exercise (Problem 1)
Prove that the Nine-Point Circle bisects any segment drawn from the orthocenter to the circumcircle.
Exercise (Problem 2)
If the orthocenter lies on the incircle, prove that the Nine-Point Circle passes through the incenter (or is related significantly).
Exercise (Problem 3)
Find the equation of the Nine-Point Circle for the triangle with vertices .