The result as follows from special configuration of merge [Nine Circle Theorem](https://mathworld.wolfram.com/NineCirclesTheorem.html) and [Eight Circle theorem](https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_t%C3%A1m_%C4%91%C6%B0%E1%BB%9Dng_tr%C3%B2n) but it is new:

Problem: Let three circle $(A)$, $(B)$, $(C)$ , let $A_c$ be arbitrary point in the circle $(C)$. Construct the circle $(O_1)$ through $A_c$ tangent to $(C)$ and $(B)$, denote $(O_1)$ tangent to $(B)$ at point $A_b$; Construct the circle $(O_2)$ through $A_b$ tangent to $(B)$ and $(A)$, denote $(O_2)$ tangent to $(A)$ at point $B_a$; Construct the circle $(O_3)$ through  $B_a$ tangent to $(A)$ and $(C)$, denote $(O_3)$ tangent to $(C)$ at point $B_c$; Construct the circle $(O_4)$ through $B_c$ tangent to $(C)$ and $(B)$, denote $(O_4)$ tangent to $(B)$ at point $C_b$; Construct the circle $(O_5)$ through  $C_b$ tangent to $(B)$ and $(A)$, $(O_5)$ tangent to $(A)$
at point $C_a$; Construct the circle $(O_6)$ through $C_a$ tangent to $(A)$ and $(C)$, then

[![enter image description here][1]][1]
1) By the [Nine Circle Theorem](https://mathworld.wolfram.com/NineCirclesTheorem.html) $O_6$ also tangent to $(O_1)$ but it is is the point of tangency is $A_c$. 
2) Six points $A_c, A_b, B_a, B_c, C_b, C_a $ lie on a circle.

**Remarks**: This result 2. show that this configuration is special configuration of [Eight Circle theorem](https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_t%C3%A1m_%C4%91%C6%B0%E1%BB%9Dng_tr%C3%B2n), when two big circles coincide, please see [Eight Circle theorem](https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_t%C3%A1m_%C4%91%C6%B0%E1%BB%9Dng_tr%C3%B2n)

3) Three lines $O_1O_4, O_2O_5, O_3O_6$ are concurrent;
4) Three lines $AO_1, BO_3, CO_5$ are concurrent
5) Three line $AO_2, BO_4, CO_6$ are concurrent
6) The points of concurrence in (3), (4), (5) are collinear.

>> **Question**: I am looking for a proof the result above

  [1]: https://i.sstatic.net/J9Xn8.png