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 $O$.

Drawing to the problem

  • 2
    $\begingroup$ There is no question in your problem. $\endgroup$ – Narasimham Mar 21 '16 at 14:59
  • $\begingroup$ Dear @Narasimham I eidted $\endgroup$ – Oai Thanh Đào Mar 22 '16 at 8:56
  • $\begingroup$ Where did you get it from? What methods are you supposed to use? $\endgroup$ – evgeny Mar 22 '16 at 12:18
  • $\begingroup$ Dear @se0808 , I thank to You very much for help me edited $\endgroup$ – Oai Thanh Đào Mar 22 '16 at 12:19
  • $\begingroup$ In MO she said "I am an electrical system engineer I live in Viet Nam. I am not a Mathematician." $\endgroup$ – Narasimham Mar 22 '16 at 12:59

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