# A chain of six circles associated with a conic

I am looking for a solution of the following problem:

Let $A_1, ..., A_6$ and $B_1, ..., B_6$ be 12 points lying on a conic, and suppose that for $i=1, ..., 5$ through $A_i, A_{i+1}, B_{i+1}, B_i$ passes a circle $(O_i)$. Then through $A_6, B_6, A_1, B_1$ as well passes a circle $(O_6)$. Let $P_1, P_4$ be intersection points of $(O_1)$ and $(O_4)$; the same for $P_2, P_5$ and $P_3, P_6$. Show that:

1. Three lines $O_1O_4$, $O_2O_5$, and $O_3O_6$ have a common point $O$.

2. Six points $P_1, ..., P_6$ lie on a circle with center in $O$.

• There is no question in your problem. – Narasimham Mar 21 '16 at 14:59
• Dear @Narasimham I eidted – Oai Thanh Đào Mar 22 '16 at 8:56
• Where did you get it from? What methods are you supposed to use? – evgeny Mar 22 '16 at 12:18
• Dear @se0808 , I thank to You very much for help me edited – Oai Thanh Đào Mar 22 '16 at 12:19
• In MO she said "I am an electrical system engineer I live in Viet Nam. I am not a Mathematician." – Narasimham Mar 22 '16 at 12:59