+750 In parallelogram EFGH, let M be the point on ¯EF such that FM:ME=1:1, and let N be the point on ¯EH such that HN:NE=1:1. Line segments ¯FH and ¯GM intersect at P, and line segments ¯FH and GN intersect at Q. Find PQ/FH.
+130477 
Triangle HQN is similar to triangle FQG Triangle HPG is similar to triangle FPM
HQ / NH = FQ / GF PH / HG = PF / FM
HQ / 2 = FQ / 4 PH / 4 = PF / 2
HQ / FQ = 2/4 = 1/2 PF / PH = 2/4 =1/2
HQ = (1/3)FH
PF = (1/3) FH
PQ = (FH - PF - HQ) = (FH - (1/3)FH - (1/3)FH) = (1/3)FH
So
PQ / FH = 1/3
