# 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.

## Proof

Let the altitudes of $\triangle ABC$ be $AD$, $BE$ and $CF$.

Let $H$ be the orthocenter of $\triangle ABC$.

Let $X$, $Y$ and $Z$ bisect $AH$, $BH$ and $CH$, respectively.

Let $A_m$, $B_m$, and $C_m$ bisect $BC$, $AC$, and $AB$, respectively.

\(\ds \triangle AHC\) | \(\sim\) | \(\ds \triangle B_mZC\) | Triangles with One Equal Angle and Two Sides Proportional are Similar | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds AXHD\) | \(\parallel\) | \(\ds B_mZ\) | |||||||||||

\(\ds \triangle AHB\) | \(\sim\) | \(\ds \triangle C_mYB\) | Triangles with One Equal Angle and Two Sides Proportional are Similar | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds AXHD\) | \(\parallel\) | \(\ds C_mY\) | |||||||||||

\(\ds B_mZ\) | \(\parallel\) | \(\ds C_mY\) | Parallelism is Transitive Relation |

\(\ds \triangle YZH\) | \(\sim\) | \(\ds \triangle BCH\) | Triangles with One Equal Angle and Two Sides Proportional are Similar | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds YZ\) | \(\parallel\) | \(\ds BDC\) | |||||||||||

\(\ds \triangle AC_mB_m\) | \(\sim\) | \(\ds \triangle ABC\) | Triangles with One Equal Angle and Two Sides Proportional are Similar | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds C_mB_m\) | \(\parallel\) | \(\ds BDC\) | |||||||||||

\(\ds C_mB_m\) | \(\parallel\) | \(\ds YZ\) | Parallelism is Transitive Relation |

By Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel:

- $\Box B_mZYC_m$ is a parallelogram.

\(\ds AXHD\) | \(\perp\) | \(\ds BC\) | Definition of Altitude of Triangle | |||||||||||

\(\ds ZY\) | \(\parallel\) | \(\ds BC\) | above | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds ZY\) | \(\perp\) | \(\ds AXHD\) | |||||||||||

\(\ds B_mZ\) | \(\parallel\) | \(\ds AXHD\) | above | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds B_mZ\) | \(\perp\) | \(\ds ZY\) | |||||||||||

\(\ds \angle B_mZY\) | \(=\) | \(\ds 90 \degrees\) | Definition of Perpendicular |

By Parallelogram with One Right Angle is Rectangle:

- $\Box B_mZYC_m$ is a rectangle.

\(\ds B_mY\) | \(=\) | \(\ds C_mZ\) | Diagonals of Rectangle are Equal |

Draw the circle with $B_mY$ as diameter.

- $Z$ and $C_m$ are also on the circle.

Since $ZC_m = Y B_m$:

Using diameter $B_mY$, by Thales' Theorem/Converse:

- $E$ is on the circle.

Using diameter $C_mZ$, by Thales' Theorem/Converse:

- $F$ is on the circle.

Using diameter $A_mX$, by Thales' Theorem/Converse:

- $D$ is on the circle.

The result follows.

$\blacksquare$

## 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**