Rather than trying to cross every line segment just once, label the vertices of this network with values of 1 through 12 such that the sum of the intersections which lie on the perimeter of the internal quadrilaterals is the same.

With a multitude of solutions abound these following impositions are to be applied separately/individually as well:
1. L4=L5 and 3*L1 = 4*L3
2. L4=L5 and 2*L1 = 3*L3
3. L4=L5 and L3<10
where L1=A+B+C+DAA L2=E+F+G+H+IAA
AAAAL3=J+K+LAAAAAAL4=A+E+JAA
AAAAL5=D+I+L.

Each impost has 2 solutions which are not reflections. How many can you find?

(In reply to Questions by Dej Mar)

Bother!  I trust this dissertation provides what you seek.
I note there is something else that was not caught prior to posting.   I shall ask levik to attempt to rectify the text for posterity.

Vertices should be interpreted as intersections. 
and
 "sum of the vertices on the perimeter"  should be read as:
"sum of the intersections which lie on the perimeter ...."  so the squares will have 4 values while the rectangles will have 5.

The letters A-L (12) label the intersection to which they are closest.

The Lx values are Line totals (would have been better to have used a different coding).

Lastly, and most importantly what no-one caught me up on was that A-L are distinct values 1-12.

Posted by brianjn on 2012-04-11 03:55:56