If $a,b,c,a_1,b_1,c_1$ are X-coordinate of $A,B,C,A_1,B_1,C_1$ then $AB \parallel A_1B_1 \to ab=a_1b_1, BC \parallel B_1C_1 \to bc=b_1c_1$ so $\frac{a}{c}=\frac{a_1}{c_1} \to ac_1=a_1c \to AC_1 \parallel A_1C$

Alternatively, one could use Pascal's theorem on $ABCA_1B_1C_1$, but I suppose that's overkill