# Nine Point Circle Theorem

## Theorem

Let $\triangle ABC$ be a triangle.

These $9$ points:

the feet of the altitudes of $\triangle ABC$
the midpoints of the sides of $\triangle ABC$
the midpoints of the lines from the vertices of $\triangle ABC$ to the orthocenter $H$ of $\triangle ABC$

all lie on the circumference of a circle.

The center $M$ lies on the Euler line of $\triangle ABC$, at the midpoint between the orthocenter $H$ and the circumcenter $O$. This circle is known as the Feuerbach circle, or the nine point circle.

## Also known as

This theorem is also known as the ninepoint circle theorem or the nine-point circle theorem -- there is little consistency in the literature.

## Historical Note

The Nine Point Circle Theorem was proved in $1820$ by Jean-Victor Poncelet and Charles Julien Brianchon.

In $1822$, Karl Wilhelm Feuerbach proved that the nine point circle was tangent to the incircle and all $3$ excircles.

Hence this circle is often referred to as the Feuerbach circle.