In given figure, \(l\) is parallel to \(m\) and line segments \(WX\),\(YZ\) and \(EF\) concurrent at point \(P\). Jusity that \(\frac{WE}{XF} = \frac{WY}{XZ} = \frac{YE}{FZ}\).
In \(\Delta WPY\) and \(\Delta XPZ\),
\(\angle WPY = \angle XPZ\) (vertically opposite angles)
\(\angle PWY = \angle PXZ\) (alternate angles)
Therefore, \(\Delta WPY \sim \Delta XPZ\) (by \(AA\) similarity criterion)
\(\frac{WP}{PX} = \frac{WY}{XZ} =\) - - - - (i)
In \(\Delta WPE\) and \(\Delta XPF\),
\(\angle WPE = \angle XPF\) (vertically opposite angles)
\(\angle PWE = \angle PXF\) (alternate angles)
Therefore, \(\Delta WPE \sim \Delta XPF\) (by \(AA\) similarity criterion)
\(\frac{WP}{PX} = \frac{WE}{XF} = \) - - - - (ii)
In \(\Delta PEY\) and \(\Delta PFZ\),
\(\angle EPY = \angle FPZ\) (vertically opposite angles)
\(\angle PYE = \angle PZF\) (alternate angles)
Therefore, \(\Delta PEY \sim \Delta PFZ\) (by \(AA\) similarity criterion)
\(\frac{PE}{PF} = \frac{PY}{PZ} =\) - - - - (iii)
From equation, we get the result.