# Possible triangle center associated with Apollonius circle of excircles

When I was playing around with Geogebra, I personally found a possible triangle center, but I'm not 100 % sure if my personal conjecture is true.

Consider the following configuration:

Let $$E_A$$, $$E_B$$, $$E_C$$ be the excircles of a given triangle $$ABC$$. Let $$S$$ be the Apollonius circle internally tangent to the three excircles. Denote the point of tangency of $$E_A$$ with $$S$$ by $$T_A$$. Define $$T_B$$ and $$T_C$$ similarly. Draw the circle $$S_A$$ that is tangent to $$AB$$ and $$CA$$ and internally tangent to $$S$$, other than the three excircles. Denote the point of tangency of $$S_A$$ with $$S$$ by $$U_A$$. Define $$S_B$$, $$S_C$$, $$U_B$$, and $$U_C$$ cyclically.

It seems that the three lines $$T_A U_A$$, $$T_B U_B$$, $$T_C U_C$$ are concurrent.

Question: Are these lines actually concurrent or not? If it's true and is an already known result, does this theorem or point have a name?

Edit
A demonstration of this proposition on Geogebra applet is available here.

Let $$I_A$$, $$I_B$$, $$I_C$$, $$O_A$$, $$O_B$$, and $$O_C$$ be the centers of $$E_A$$, $$E_B$$, $$E_C$$, $$S_A$$, $$S_B$$, and $$S_C,$$ respectively. Let $$I$$ be the incircle of $$\triangle{ABC}$$. Denote by $$B_A$$ and $$A_B$$ the intersections of the ray $$\overrightarrow{AB}$$ with $$S$$ and the ray $$\overrightarrow{BA}$$ with $$S$$, respectively. $$A_B$$, $$A$$, $$B$$, $$B_A$$ lie on a line in this order. Define $$C_B$$, $$B_C$$, $$A_C$$, $$C_A$$ cyclically.

1. $$T_A U_A$$, $$T_B U_B$$, $$T_C U_C$$ are concurrent
$$\iff \frac{\vert \overline{U_C T_B} \vert}{\vert \overline{T_B U_A} \vert} \cdot \frac{\vert \overline{U_A T_C} \vert}{\vert \overline{T_C U_B} \vert} \cdot \frac{ \vert \overline{U_B T_A} \vert}{\vert \overline{T_A U_C} \vert}=1$$ (Ceva's theorem for chords)
2. $$BC$$ and $$B_A C_A$$ are parallel (parallel tangent theorem). $$S$$ is a Tucker circle centered at $$X_{13323}$$ of the triangle bounded by $$B_A C_A$$, $$C_B A_B$$, and $$A_C B_C$$. See also Darij Grinberg and Paul Yiu, "The Apollonius Circle as a Tucker Circle".
3. $$T_B$$, $$T_C$$, and the exsimilicenter of $$E_B$$ and $$E_C$$ are collinear. This is a special case of Monge's theorem.
4. $$A T_A$$, $$B T_B$$, and $$C T_C$$ meet at the exsimilicenter of $$S$$ and $$I$$, known as Apollonius point $$X_{181}$$. Similarly, $$A U_A$$, $$B U_B$$, $$C U_C$$ meet at the insimilicenter of $$S$$ and $$I$$, which is $$X_{1682}$$. These are consequences of Monge's theorem as well.
5. $$A_B A_C$$, $$B_A C_A$$, and $$T_B T_C$$ are concurrent.
6. $$I_B$$, $$I_C$$, the incenter of $$\triangle{A_B B_C C_B}$$ and the incenter of $$\triangle{A_C B_C C_B}$$ are collinear. Similarly, $$I_B$$, $$O_C$$, the incenter of $$\triangle{C_B A_C C_A}$$ are collinear (Sawayama-Thébault's theorem).
7. $$A_C$$, $$U_A$$, the point of contact of $$S_A$$ with $$A A_C$$, the incenter of $$\triangle{A A_B A_C}$$, and the incenter of $$\triangle{A_C B_A A_B}$$ are concyclic (attributed to Yūzaburō Sawayama).
8. If $$E_A$$ is not tangent to $$BC$$, then the proposition does not hold generally. Even if we assume that $$BC \parallel B_A C_A$$, $$CA \parallel C_B A_B$$, and $$AB \parallel A_C B_C$$, the proposition is generally not true.

I found another way to construct this point: Let $$V_1=T_A A_B \cap T_C C_B$$, $$V_2=T_C C_B \cap T_B B_C$$, $$V_3=T_B B_C \cap T_A A_C$$, $$V_4=T_A A_C \cap T_C C_A$$, $$V_5=T_C C_A \cap T_B B_A$$, and $$V_6=T_B B_A \cap T_A A_B$$. The three lines $$V_1 V_4$$, $$V_2 V_5$$, $$V_3 V_6$$ meet at this point.

I could not find this center in Kimberling's Encyclopedia of Triangle Centers, at least in the list from $$X_1$$ to $$X_{47627}$$.

Let $$a$$, $$b$$, $$c$$ be the lengths of the sides $$\overline{BC}$$, $$\overline{CA}$$, $$\overline{AB}$$, respectively. Then, $$X=(f(a,b,c):f(b,c,a):f(c,a,b)),$$ where $$f(a,b,c)=a^2(b^2+c^2+a(b+c))(a^4(b+c)^2-bc(b+c)^2(b^2+c^2)+a^3(b+c)(b^2+bc+c^2)-a(b^2-c^2)(b^3-c^3)-a^2(b^4+c^4-bc(b^2+6bc+c^2)))$$
This triangle center was not listed in the Encyclopedia of Triangle Centers. Now it is $$X_{50032}$$.