# 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 $O$.

This circle is known as the Feuerbach circle, or the nine point circle.

## Proof

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## 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.

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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $9$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $9$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**nine-point circle** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**nine-point circle** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**nine-point circle**