Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A_1, A_2$. Let a conic through $C, C', A, A'$ and meets the cubic again at $B_1, B_2$. Let a conic through $A, A', B, B'$ and meets the cubic again at $C_1, C_2$. Then six points $A_1,A_2, B_1, B_2, C_1, C_2$ lie on a conic. In this case if we let $A_1, B_1, C_1$ are collinear then $A_2, B_2, C_2$ are collinear.

**Special case 1, The cubis are three lines.** Let six points $A, A', B, B', C, C'$ lie on a conic.
Let a conic through $B, B', C, C'$ and meets line $AA'$ at $A_1, A_2$
Let a conic through $C, C', A, A'$ and meets line $BB'$ at $B_1, B_2$
Let a conic through $A, A', B, B'$ and meets line $CC'$ at $C_1, C_2$.
Then six points $A_1,A_2, B_1, B_2, C_1, C_2$ lie on a conic. In this case if $A_1, B_1, C_1$ are collinear then $A_2, B_2, C_2$ are collinear.

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**Special case 2, A conic and a cubic:** Let a conic meets a cubic at six points $A, B, C, A', B', C'$. Let $AB', A'B$ cut the cubic at $C_1, C_2$ respectively; Let $BC', B'C$ cut the cubic again at $A_1, A_2$; Let $AC', A'C$ cut the cubic at $B_2, B_1$ then $A_1, B_1, C_1$ are collinear and $A_2, B_2, C_2$ are collinear.

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**Special case 3, The conics are lines:** Let two lines meet a cubic at $A, B, C$; and $A', B', C'$ respectively. Let $AB', A'B$ cut the cubic at $C_1, C_2$ respectively; Let $BC', B'C$ cut the cubic again at $A_1, A_2$; Let $AC', A'C$ cut the cubic at $B_2, B_1$ then $A_1, B_1, C_1$ are collinear and $A_2, B_2, C_2$ are collinear.

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**Special case 4, The cubic are a line and a conic:** Let the cubic is a conic ($\gamma$) through four points $A, A', B, B'$ and line $CC'$ on the plane). Let points $A, B, C, A', B', C'$ lie on another conic. Let $AB', A'B$ meet $CC'$ at $C_1, C_2$; let $AC', A'C, BC', B'C$ meet conic $\gamma$ at $B_2, B_1, A_1, A_2$. Then six points $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a conic.