Incidence in Feuerbach's Theorem: What Is It About?
A Mathematical Droodle


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Copyright © 1996-2018 Alexander Bogomolny


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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 Fa, Fb, Fc (for the excircles) and Fi for the incircle. Then there are two cases of incidence:

  1. The three lines AFa, BFb and CFc are concurrent in point F, say.

  2. The four points N, F, I, Fi 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 Fi. The line goes under a generic reference L(1, 5).

Nine Point Circle

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Copyright © 1996-2018 Alexander Bogomolny

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