TITLE: Geometry. Parallelogram.
QUESTION [6 upvotes]: Let $A$, $B$, $C$, $D$ be $4$ fixed points on line $l$. Through $A$ and $B$ pass two arbitrary parallel lines, and two arbitrary parallel lines pass through $C$ and $D$. These $4$ lines form a parallelogram. Prove, that the diagonals of this parallelogram cross the line $l$ in two fixed points. Tried Menelaus' Theorem and  the Intercept theorem but couldn't work it out. How to solve this?

REPLY [2 votes]: Use coordinate geometry

Calculate the coordinates of the vertices V1 and V2 and then fit the equation of the diagonal. The intercept of the diagonal with the y axis is Y = [b(a-d) + a(c-b)] / [a - d + c - b], provided m not equal to n and provided (a - d + c -b) not equal 0. The first is the condition that the two sets of lines are not parallel and the second that the diagonal is not parallel to the y axis.
So, the y intercept is independant of the slope of the two sets of parallel lines.