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.

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 is also known as:

the ninepoint circle theorem

the nine-point circle theorem.

There is little consistency in the literature.

Also see

Feuerbach's Theorem

Historical Note

The 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 (C.J. Brianchon and J.V. Poncelet, 1820; K.W. Feuerbach 1822)
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): nine-point circle