### Incidence in Feuerbach's Theorem: What Is It About?

A Mathematical Droodle

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According to Feuerbach's theorem, the 9-point circle of a triangle touches its incircle and the three excircles. The point of tangency of the incircle and the 9-point circle is known as the *Feuerbach point* N. Let the base triangle be ABC, and denote the points of tangency in question F_{a}, F_{b}, F_{c} (for the excircles) and F_{i} for the incircle. Then there are two cases of incidence:

The three lines AF

_{a}, BF_{b}and CF_{c}are concurrent in point F, say.The four points N, F, I, F

_{i}are concurrent.

Neither the point in #1, nor the line in #2, have an official appellation. The point is X(12) in Kimberling's Encyclopedia of Triangle Centers, a harmonic conjugate of X(11), the Feuerbach Point F_{i}. The line goes under a generic reference

### Nine Point Circle

- Nine Point Circle: an Elementary Proof
- Feuerbach's Theorem
- Feuerbach's Theorem: a Proof
- Four 9-Point Circles in a Quadrilateral
- Four Triangles, One Circle
- Hart Circle
- Incidence in Feuerbach's Theorem
- Six Point Circle
- Nine Point Circle
- 6 to 9 Point Circle
- Six Concyclic Points II
- Bevan's Point and Theorem
- Another Property of the 9-Point Circle
- Concurrence of Ten Nine-Point Circles
- Garcia-Feuerbach Collinearity
- Nine Point Center in Square

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Copyright © 1996-2018 Alexander Bogomolny

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