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
1/x + 1/y = 1/z