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
The third Fontené theorem states that the pedal circle of a point $P$ is tangent to the nine point circle iff $P$ and its isogonal conjugate $P'$ lie on a line through the circumcenter.
Now we take $P$ to be an incenter or excenter of $\triangle ABC$.
By Isogonal Conjugate of an Incenter or Excenter is Itself, we have $P = P'$.
Therefore $P$ and $P'$ lie on a line through the circumcenter.
By the third Fontené theorem the pedal circle of $P$ is tangent to the nine point circle.
The pedal circle of incenter is incircle.
The pedal circle of excenter is incircle.
Hence the result.
$\blacksquare$
Also see
Source of Name
This entry was named for Karl Wilhelm Feuerbach.
Historical Note
Feuerbach's Theorem 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 (Universitext): Exercise III.64
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): nine-point circle
- 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
- Weisstein, Eric W. "Fontené Theorems." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FonteneTheorems.html
- Weisstein, Eric W. "Feuerbach's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FeuerbachsTheorem.html