# New circles associated with the Miquel theorem configuration

Consider the Miquel six circle theorem configuration

Let $A, B, C, D$ are concyclic, $A_1, B_1, C_1, D_1$ are concyclic, and $A, A_1, B_1, ,B$; $B, B_1, C_1 C$, $C, C_1, D_1, D$; $D, D_1, A_1, A$ concyclic. Let $P_1, P_2 = (ACC_1) \cap (DBB_1)$ and $P_4, P_5 = (CAA_1) \cap (BDD_1)$. Then show that $P_1, P_2, P_3, P_4$ are concyclic and $(P_1P_2P_3P_4)$$(CC_1D_1D)$ and $(AA_1B_1B)$ are coaxial.

• Nice setup! Where does this problem come from? What proof techniques for Miquel's theorem are you familiar with, and to what extend can they be applied to the problem at hand? – MvG Mar 15 '16 at 9:31
• @MvG , I found this problem when I draw Miquel theorem by using Geogebra sofware. I am looking for a proof. – Oai Thanh Đào Mar 15 '16 at 14:18
• – Oai Thanh Đào Mar 15 '16 at 14:49