In Δ FDC and Δ FBA,

∠FDC = ∠FBA [As DC || AB]

∠DFC = ∠BFA [common angle]

Hence, ∆FDC ~ ∆FBA by AA criterion for similarity.

So, we have DC/AB = DF/BF z/x = DF/BF …. (1)

In Δ BDC and Δ BFE, ∠BDC = ∠BFE [As DC || FE]

∠DBC = ∠FBE [Common angle]

Hence, ∆BDC ~ ∆BFE by AA criterion for similarity.

So, we have BD/BF = z/y ….. (2)

Now, adding (1) and (2), we get

BD/BF + DF/BF = z/y + z/x

1 = z/y + z/x

Thus, 1/z = 1/x + 1/y

**Answer**

1/x + 1/y = 1/z