Nine Point Circle Theorem

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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


Proof


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


Historical Note

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


Sources