Consider
a/b = c/d = e/f = k
So we get
a = bk, c = dk, e = fk
(i) LHS = \[({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~+\text{ }{{e}^{2}})\]
We can write it as
\[=\text{ }({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }({{b}^{2}}{{k}^{2}}~+\text{ }{{d}^{2}}{{k}^{2}}~+\text{ }{{f}^{2}}{{k}^{2}})\]
Taking out the common terms
\[=\text{ }({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }{{k}^{2}}~({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\]
So we get
\[=\text{ }{{k}^{2}}~({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\]
RHS \[=\text{ }{{\left( ab\text{ }+\text{ }cd\text{ }+\text{ }ef \right)}^{2}}\]
We can write it as
\[=\text{ }{{\left( b.\text{ }kb\text{ }+\text{ }dk.\text{ }d\text{ }+\text{ }fk.\text{ }f \right)}^{2}}\]
So we get
\[=\text{ }(k{{b}^{2}}~+\text{ }k{{d}^{2}}~+\text{ }k{{f}^{2}})\]
Taking out common terms
\[=\text{ }{{k}^{2}}~{{({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})}^{2}}\]
Therefore, LHS = RHS.
Therefore, LHS = RHS