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I am looking for proof of a problem as follows:

Let three points $A, B, C$ are collinear. Let three lines $a, a_1, a_2 $ through $A$, three lines $b, b_1, b_2 $ through $B$ three lines $c, c_1,c_2$ through $C$. Such that $a_1 \parallel b_2; b_1 \parallel c_2, c_1 \parallel a_2 $. Let $A_1=a \cap b_1, A_2=a\cap c_2$ $B_1=b\cap c_1, B_2=b \cap a_2$, $C_1=c \cap a_1, C_2=c \cap b_2$. Then show that midpoints of $A_1C_2, B_1A_2, C_1B_2$ are collinear.

The problem on configuration of the Pappus theorem

See also: Elementary proof of a triangular grid lemma enter image description here

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Let us fix all direction of lines and move the point $C$ along the black line. We see that all points linearly depend from the position of $C$, therefore it is sufficient to check the statement for point $C=A$, $B$ and the infinity. This is not very hard, for example we see, that direction of the segment connecting midpoints of $A_1C_2$ and $B_2C_1$ does not change.

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