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I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane).

This is a generalization of the last my previous question in Three chains of six circles. (Noting that, in this configuration: $P'_1P_1P_4P'_4$ don't lie on a circle) I am looking for a proof of the conjecture as following:

Conjecture:

In the Möbius plane, let a chain of six circles $(C_1)$, $(C_2)$, $(C_3)$, $(C_4)$, $(C_5)$, $(C_6)$, such that two neighbors circles $(C_i)$ meets $(C_{i+1})$ at two points $P_i, P'_i$ where $i=1, 2, 3, 4, 5, 6$; Such that six points $P_1, P_2, P_3, P_4, P_5, P_6$ lie on a circle.

Let $A_1$ be a point on the circle $(C_6)$, let the circle $(A_1P_1P_2)$ meets the circle $(C_2)$ at $A_2$, Let the circle $(A_2P'_3P'_4)$ meets the circle $(C_4)$ at $A_3$, Let the circle $(A_3P_5P_6)$ meets the circle $(C_6)$ at $A_4$, Let the circle $(A_4P'_1P'_2)$ meets the circle $(C_2)$ at $A_5$, Let the circle $(A_5P_3P_4)$ meets the circle $(C_4)$ at $A_6$. Then show that:

  • Four points $A_6, P_5, P_6, A_1$ lie on a circle.

  • Six points $A_1, A_2, A_3, A_4, A_5, A_6$ lie on a circle

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    $\begingroup$ @Anton: I disagree strongly with this attitude. The problem "for us" is e.g. realizing the outer circle as a principal homogeneous space with respect to the action of the inner circle, thereby making this conjecture "obvious" (I have no idea whether this will work - that's why it is a problem). I do agree that in its present form this question is better suited for MSE. $\endgroup$ – Franz Lemmermeyer Apr 3 '16 at 11:07
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    $\begingroup$ @OaiThanhĐào you have nice problems, but they are not good for MO. $\endgroup$ – Anton Petrunin Apr 3 '16 at 12:15
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    $\begingroup$ @AntonPetrunin The conjecture true for Möbius plane $\endgroup$ – Oai Thanh Đào Apr 3 '16 at 12:22
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    $\begingroup$ Nice questions, but not for this website, IMHO. $\endgroup$ – Arseniy Akopyan Apr 4 '16 at 15:47
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    $\begingroup$ Because it was not a classical geometry question. $\endgroup$ – Arseniy Akopyan Apr 5 '16 at 18:07