Let $A,B,C,C_1,B_1A_1$ lies on a conic. Let $A_2=BC_1 \cap B_1C, B_2=AC_1 \cap A_1C; C_2=AB_1 \cap A_1B$. Let $A_3,B_3,C_3$ lie on sidelines $BC,CA,AB$ such that $A_3,B_3,C_3$ are collinear. Let $A_4=A_3A_2 \cap B_1C_1, B_4=B_3B_2 \cap A_1C_1; C_4=A_3B_3 \cap A_1B_1$ . Then $A_4,B_4,C_4$ are collinear.
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