For these lines to intersect, they must not be vertical neither horizontal. So it seems by inductive reasoning. Thus they must be oblique or slanted parallel lines.
With a quick sketch, the attachment of this post is what I have in mind.
Hey guys, I have an idea on how to do this problem for certain cases - does anyone know how to show this for a general case (since thats what its asking for I think):
"If we have 4 points A,B,C, D which are given on a straight line (in that order), show how to construct a pair of paralllel lines through 'A' and 'B' and another pair of parallel lines through 'C' and 'D' so that the 4 lines intersect in a square"
If the resulting figure is to be a square, then the A,B,C,D must be equally spaced.
Get the midpoint of BC. Call it M.
With M as center, draw semicircles with A and B. (A and D must be on the larger semicircle, while B and C must be on the smaller semicircle.)
Draw a line through M that is perpendicular to the original line. This line will bisect the two semicircles.
Draw a line through A that will pass through the the point of intersection of the larger semicircle and the perpendicular line from M.
Do that too from D.
Draw a line through B that will pass through the the point of intersection of the smaller semicircle and the perpendicular line from M.
Do that too from C.
What is the figure formed now by those four lines?
1. It doesn't matter how the 4 points are placed on the line segment. If the order is A, B, C, D then you always can construct a square.
2. The vertices of the square are located on semi-circles over AC, AD, BC, BD.
3. I haven't found the most important property to place one vertix definitely on it's semi-circle.