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

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Also see

• Nine Point Circle Theorem

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
• 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