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


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.

9PointCircleLabels.png
\(\ds \triangle AHC\) \(\sim\) \(\ds \triangle B_mZC\) 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.

By Thales' Theorem/Converse:

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

Since $ZC_m = Y B_m$:

$ZC_m$ is a diameter of the circle.

mutatis mutandis:

$XA_m$ 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

The nine point circle theorem is also known as:

the ninepoint circle theorem
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

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