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

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 L(1, 5).

Nine Point Circle

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