Definition:Feuerbach Circle

Definition

Let $\triangle ABC$ be a triangle.

The Feuerbach circle of $\triangle ABC$ is the circle which passes through each of the $9$ 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 of $\triangle ABC$.

Also known as

The Feuerbach circle of a triangle is also known as the nine point circle or nine-point circle.

Also see

• Nine Point Circle Theorem
• Feuerbach's Theorem

• Results about Feuerbach circles can be found here.

Source of Name

This entry was named for Karl Wilhelm Feuerbach.

Historical Note

Although named after Karl Wilhelm Feuerbach, the Feuerbach Circle was actually discovered in $1820$ by Jean-Victor Poncelet and Charles Julien Brianchon.

It bears Feuerbach's name as the result of Feuerbach's Theorem, which extends the known properties of the circle to include the tangencies to the incircle and excircles.

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