Question

In figure, l || m and line segments AB, CD and EF are concurrent at point P. Prove that `(AE)/(BF) = (AC)/(BD) = (CE)/(FD)`.

Answer 1

Given l || m and line segments AB, CD and EF are concurrent at point P.

To prove: `("AE")/("BF") = ("AC")/("BD") = ("CE")/("FD")`

Proof: In ΔAPC and ΔBPD,

 ∠APC = ∠BPD   ...[Vertically opposite angles]

∠PAC = ∠PBD   ...[Alternative angles]

∴ ΔAPC ∼ ΔBPD   ...[By AA similarity criterion]

Then, `("AP")/("PB") = ("AC")/("BD") = ("PC")/("PD")`   ...(i)

In ΔAPE and ΔBPF,

∠APE = ∠BPF   ...[Vertically opposite angles]

∠PAE = ∠PBF   ...[Alternative angles]

∴ ΔAPE ∼ ΔBPF   ...[By AA similarity criterion]

Then, `("AP")/("PB") = ("AE")/("BF") = ("PE")/("PF")`   ...(ii)

In ΔPEC and ΔPFD,

∠EPC = ∠FPD   ...[Vertically opposite angles]

∠PCE = ∠PDF   ...[Alternative angles]

∴ ΔPEC ∼ ΔPFD  ...[By AA similarity criterion]

Then, `("PE")/("PF") = ("PC")/("PD") = ("EC")/("FD")`   ...(iii)

From equations (i), (ii) and (iii),

`("AP")/("PB") = ("AC")/("BD") = ("AE")/("BF") = ("PE")/("PF") = ("EC")/("FD")`

∴ `("AE")/("BF") = ("AC")/("BD") = ("CE")/("FD")`

Hence proved.