Question

AB//PQ//CD ,AB=X,CD=Y,PQ=Z prove that 1/X+1/Y=1/Z

Answer 1

see the diagram.

ΔABD and ΔPQD are similar, as the corresponding sides are parallel.
      x / z = BD / QD
             =>  1/x = QD /(z * BD)   ---(1)

ΔCDB and ΔPQB are similar, as corresponding sides are parallel.
     y / z = BD / BQ
             =>  1/y = BQ / (z * BD) ---(2)

Add the two equations: 
         1/x + 1/y = (BQ + QD) / (z * BD) = 1 / z