In given figure, \(l || m\) and line segments \(PQ\),\(RS\) and \(EF\) concurrent at point \(P\). Prove that \(\frac{PE}{QF} = \frac{PR}{QS} = \frac{RE}{FS}\).
In \(\Delta PPR\) and \(\Delta QPS\),
\(\angle PPR = \angle QPS\) (vertically opposite angles)
\(\angle PPR = \angle PQS\) (alternate angles)
Therefore, \(\Delta PPR \sim \Delta QPS\) (by \(AA\) similarity criterion)
\(\frac{PP}{PQ} = \frac{PR}{QS} =\) - - - - (i)
In \(\Delta PPE\) and \(\Delta QPF\),
\(\angle PPE = \angle QPF\) (vertically opposite angles)
\(\angle PPE = \angle PQF\) (alternate angles)
Therefore, \(\Delta PPE \sim \Delta QPF\) (by \(AA\) similarity criterion)
\(\frac{PP}{PQ} = \frac{PE}{QF} = \) - - - - (ii)
In \(\Delta PER\) and \(\Delta PFS\),
\(\angle EPR = \angle FPS\) (vertically opposite angles)
\(\angle PRE = \angle PSF\) (alternate angles)
Therefore, \(\Delta PER \sim \Delta PFS\) (by \(AA\) similarity criterion)
\(\frac{PE}{PF} = \frac{PR}{PS} =\) - - - - (iii)
From equation, we get the result.