If $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$, prove that

$\begin{array}{ll} \text{(i)} & (b^2 + d^2 + f^2)(a^2 + c^2 + e^2) \ & = (ab + cd + ef)^2 \\ \text{(ii)} & \dfrac{(a^3 + c^3)^2}{(b^3 + d^3)^2} = \dfrac{e^6}{f^6} \\ \text{(iii)} & \dfrac{a^2}{b^2} + \dfrac{c^2}{d^2} + \dfrac{e^2}{f^2} \ & = \dfrac{ac}{bd} + \dfrac{ce}{df} + \dfrac{ae}{bf} \\ \text{(iv)} & bdf\Big(\dfrac{a + b}{b} + \dfrac{c + d}{d} + \dfrac{e + f}{f}\Big)^3 \ & = 27(a + b)(c + d)(e + f) \end{array}$

(i) Let $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = k$

$\therefore a = bk, c = dk, e = fk.$

$\text{L.H.S.} = (b^2 + d^2 + f^2)(a^2 + c^2 + e^2) \\[0.5em] = (b^2 + d^2 + f^2)(b^2k^2 + d^2k^2 + f^2k^2) \\[0.5em] = k^2(b^2 + d^2 + f^2)(b^2 + d^2 + f^2) \\[0.5em] = k^2(b^2 + d^2 + f^2)^2. \\[1em] \text{R.H.S.} = (ab + cd + ef)^2 \\[0.5em] = (bk.b + dk.d + fk .f)^2 \\[0.5em] = (b^2k + d^2k + f^2k)^2 \\[0.5em] = k^2(b^2 + d^2 + f^2).$

Since, L.H.S. = R.H.S. hence proved that, (b² + d² + f²)(a² + c² + e²) = (ab + cd + ef)².

(ii) Let $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = k$

$\therefore a = bk, c = dk, e = fk.$

$\text{L.H.S.} = \dfrac{(a^3 + c^3)^2}{(b^3 + d^3)^2} \\[1em] = \dfrac{(b^3k^3 + d^3k^3)^2}{(b^3 + d^3)^2} \\[1em] = \dfrac{k^6(b^3 + d^3)^2}{(b^3 + d^3)^2} \\[1em] = k^6 \\[1em] \text{R.H.S.} = \dfrac{e^6}{f^6} \\[1em] = \dfrac{f^6k^6}{f^6} \\[1em] = k^6$

Since, L.H.S. = R.H.S. Hence proved.

(iii) Let $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = k$

$\therefore a = bk, c = dk, e = fk.$

$\text{L.H.S.} = \dfrac{a^2}{b^2} + \dfrac{c^2}{d^2} + \dfrac{e^2}{f^2} \\[1em] = \dfrac{b^2k^2}{b^2} + \dfrac{d^2k^2}{d^2} + \dfrac{f^2k^2}{f^2} \\[1em] = k^2 + k^2 + k^2 \\[1em] = 3k^2 \\[1em] \text{R.H.S.} = \dfrac{ac}{bd} + \dfrac{ce}{df} + \dfrac{ae}{bf} \\[1em] = \dfrac{(bk)dk}{bd} + \dfrac{(dk)fk}{df} + \dfrac{(bk)fk}{bf} \\[1em] = k^2 + k^2 + k^2 \\[1em] = 3k^2$

Since, L.H.S. = R.H.S. Hence proved.

(iv) Let $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = k$

$\therefore a = bk, c = dk, e = fk.$

$\text{L.H.S.} = bdf\Big(\dfrac{a + b}{b} + \dfrac{c + d}{d} + \dfrac{e + f}{f}\Big)^3 \\[1em] = bdf\Big(\dfrac{bk + b}{b} + \dfrac{dk + d}{d} + \dfrac{fk + f}{f}\Big)^3 \\[1em] = bdf(k + 1 + k + 1 + k + 1)^3 \\[1em] = bdf(3k + 3)^3 \\[1em] = bdf(3)^3(k + 1)^3 \\[1em] = 27bdf(k + 1)^3. \\[1em] \text{R.H.S.} = 27(a + b)(c + d)(e + f) \\[1em] = 27(bk + b)(dk + d)(fk + f) \\[1em] = 27b(k + 1)d(k + 1)f(k + 1) \\[1em] = 27bdf(k + 1)^3.$

Since, L.H.S. = R.H.S. Hence proved.