To prove :
Fix points $A,C$ and set point $B$ to be the midpoint of segment $AC$. Fix point $Y$ (anywhere) and consider an arbitrary point $X$ on line $YB$. If $P$ and $Q$ are the intersection points of two lines $AY \cap CX$ , $CY \cap AX$ respectively, show that line $PQ$ is parallel to line $AC$.
I first proved converse, to see what geometry this figure has in structure.And that if trapezoid $APQC$ is given and If $Y$ and $X$ are the intersection points of the sides of the trapezoid $AP\cap CQ$ and $AQ\cap PC$ respectively, then line $YX$ passes through midpoints of segment $AC$ and $PQ$. However this didn't help much.
Im trying to solve this by looking for some formula related with a point $B$. So $PQ$ and $AC$ will be parallel when $B$ is midpoint of $AC$.
I took hours with this problem and I couldn't find a clue.
What would be the proof look like?