A Seven Circles Problem

I am looking for a proof of the problem as follows:

Let $A_1, A_2,\cdots, A_6$ lie on a circle with center $O$. Let the circle $(A_1OA_2)$ meets the circles $(A_3OA_4)$ again at $A_{23}$; the circle $(A_3OA_4)$ meets the circles $(A_5OA_6)$ again at $A_{45}$; the circle $(A_5OA_6)$ meets the circles $(A_1OA_2)$ again at $A_{61}$. Let $O_{1}$, $O_{2}$, $O_{3}$, $O_{4}$, $O_{5}$, $O_{6}$ are the center of circles $(A_1OA_2)$, $(A_2A_{23}A_3)$, $(A_3OA_4)$, $(A_4A_{45}A_5)$, $(A_5OA_6)$, $(A_6A_{61}A_1)$ respectively, then three lines $O_1O_4$, $O_2O_5$, $O_3O_6$ are concurrent.

When $A_6 \equiv A_1, A_2 \equiv A_3, A_4\equiv A_5$ the problem a bove become the Kosnita's theorem

See also:

• Seven circles theorem

• Dao's theorem on six circumcenter

• Eight circles theorem

Update:

Today, a friend let me know that, this is Theorem 1.4 in here https://www.journal-1.eu/2018/Giang-An-Kosnita.pdf