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 $O$.


 * 9PointCircle.png

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.

Also see

 * Feuerbach's Theorem