Synthetic proof of X(12) - harmonic conjugate of Feuerbach point

Question

I'm looking for the synthetic proof of this interesting fact: https://www.cut-the-knot.org/Curriculum/Geometry/FeuerbachIncidence.shtml

In words from https://faculty.evansville.edu/ck6/encyclopedia/ETC.html , X(12):

Let $F_a$, $F_b$, $F_c$ be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines $AF_a$, $BF_b$, $CF_c$ meet at X(12), the harmonic conjugate of the Feuerbach point $F_i$ with respect to the incenter I and the nine-point center N.

Although the coordinate proof is not difficult, is there a synthetic proof about such a famous point?

I think the goal $(N,I;F_i,X_{12})=-1$ is enough. But I can't find a complete quadrangle to prove that.

Answer 1

A proof copied from here:

Suppose $F'$ (i.e. $X_{12}$) is internal homothetic center of incircle $(I)$ and NPC, $(I_a), (I_b), (I_c)$ are $A$ - excircle, $B$ - excircle, $C$ - excircle of $\triangle ABC$ then it's obvious the $(IN, F_iF') = - 1$. Since $A$ is external homothetic center of $(I)$ and $(I_a),$ $F_a$ is internal homothetic center of incircle $(I_a)$ and NPC so by Monge and D'Alembert theorem, we have $A, F_a, F'$ are collinear. Similarly, we have $B, F_b, F'$ are collinear; $C, F_c, F'$ are collinear