# 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
$\Box B_mZYC_m$ 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
$\Box B_mZYC_m$ 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 $ZC_m = Y B_m$:

$ZC_m$ is a diameter of the circle.
$XA_m$ is a diameter of the circle.

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.

## 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.