Feuerbach's Theorem

Theorem

Let $\triangle ABC$ be a triangle.

The Feuerbach circle of $\triangle ABC$ is tangent to:

the incircle of $\triangle ABC$

and:

the $3$ excircles of $\triangle ABC$.



Proof

Recall the Third Fontené Theorem:

the pedal circle of a point $P$ is tangent to the nine point circle

if and only if:

$P$ and its isogonal conjugate $P^{-1}$ lie on a line through the circumcenter.


Let $P$ be either the incenter or an excenter of $\triangle ABC$.

By Isogonal Conjugate of Incenter or Excenter is Itself, we have $P = P^{-1}$.

As $P$ and $P^{-1}$ are the same point, they both trivially lie on a straight line through the circumcenter of $\triangle ABC$.


Hence by the Third Fontené Theorem the pedal circle of $P$ is tangent to the nine point circle.

Then from:

Pedal Circle of Incenter is Incircle
Pedal Circle of Excenter is Excircle

the result follows.

$\blacksquare$


Also see


Source of Name

This entry was named for Karl Wilhelm Feuerbach.


Historical Note

was proved in $1822$ by Karl Wilhelm Feuerbach.

This was two years after Jean-Victor Poncelet and Charles Julien Brianchon had demonstrated the existence of the nine point circle, now known 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
  • 2003: Michèle Audin: Geometry: Exercise $\text {III}.64$
  • 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): Feuerbach's Theorem
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): nine-point circle
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