GIVEN p = $(\FRAC{{4{\rm{xy}}}}{{{\rm{x\;}} + {\rm{\;y}}}})$

$(\Rightarrow {\rm{\;}}\frac{{\rm{p}}}{{2{\rm{x}}}} = \frac{{2{\rm{y}}}}{{{\rm{x\;}} + {\rm{\;y}}}})$ and $(\frac{{\rm{p}}}{{2{\rm{y}}}} = \frac{{2{\rm{x}}}}{{{\rm{x\;}} + {\rm{\;y}}}})$.

Applying COMPONENDO and dividendo on both, we get

$(\Rightarrow {\rm{\;}}\frac{{{\rm{p\;}} + {\rm{\;}}2{\rm{x}}}}{{{\rm{p}} - 2{\rm{x}}}} = \frac{{2{\rm{y\;}} + {\rm{\;}}\left( {{\rm{x\;}} + {\rm{\;y}}} \RIGHT)}}{{2{\rm{y}} - \left( {{\rm{x\;}} + {\rm{\;y}}} \right)}})$ and $(\frac{{{\rm{p\;}} + {\rm{\;}}2{\rm{y}}}}{{{\rm{p}} - 2{\rm{y}}}} = \frac{{2{\rm{x\;}} + {\rm{\;}}\left( {{\rm{x\;}} + {\rm{\;y}}} \right)}}{{2{\rm{x}} - \left( {{\rm{x\;}} + {\rm{\;y}}} \right)}})$

$(\Rightarrow {\rm{\;}}\frac{{{\rm{p\;}} + {\rm{\;}}2{\rm{x}}}}{{{\rm{p}} - 2{\rm{x}}}} = \frac{{{\rm{x\;}} + {\rm{\;}}3{\rm{y}}}}{{{\rm{y}} - {\rm{x}}}})$ and $(\frac{{{\rm{p\;}} + {\rm{\;}}2{\rm{y}}}}{{{\rm{p}} - 2{\rm{y}}}} = \frac{{3{\rm{x\;}} + {\rm{\;y}}}}{{{\rm{x}} - {\rm{y}}}})$

$(\begin{array}{l}\therefore {\rm{\;}}\frac{{{\rm{p\;}} + {\rm{\;}}2{\rm{x}}}}{{{\rm{p}} - 2{\rm{x}}}}{\rm{\;}} + {\rm{\;}}\frac{{{\rm{p\;}} + {\rm{\;}}2{\rm{y}}}}{{{\rm{p}} - 2{\rm{y}}}} = \frac{{{\rm{x\;}} + {\rm{\;}}3{\rm{y}}}}{{{\rm{y}} - {\rm{x}}}}{\rm{\;}} + {\rm{\;}}\frac{{3{\rm{x\;}} + {\rm{\;y}}}}{{{\rm{x}} - {\rm{y}}}}\\ = \frac{{3{\rm{x\;}} + {\rm{\;y}}}}{{{\rm{x}} - {\rm{y}}}} - \frac{{{\rm{x\;}} + {\rm{\;}}3{\rm{y}}}}{{{\rm{x}} - {\rm{y}}}} = \frac{{\left( {3{\rm{x\;}} + {\rm{\;y}}} \right) - \left( {{\rm{x\;}} + {\rm{\;}}3{\rm{y}}} \right)}}{{{\rm{x}} - {\rm{y}}}}\\ = \frac{{2{\rm{x}} - 2{\rm{y}}}}{{{\rm{x}} - {\rm{y}}}} = \frac{{2\left( {{\rm{x}} - {\rm{y}}} \right)}}{{{\rm{x}} - {\rm{y}}}} = 2\end{array})$