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If $\frac{a^{2} + c^{2}}{a b + c d} = \frac{a b + c d}{b^{2} + d^{2}}$ , then $\frac{a}{b} = \frac{c}{d}$
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\frac{a^{2} + c^{2}}{a b + c d} = \frac{a b + c d}{b^{2} + d^{2}}

, then

\frac{a}{b} = \frac{c}{d}

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\frac{a}{b} = \frac{c}{d} = \frac{e}{f}

then show that

(b^{2} + d^{2} + f^{2}) \times (a^{2} + c^{2} + e^{2}) = (a b + c d + e f)^{2}

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\sqrt{\frac{a^{2} + 5 c^{2}}{b^{2} + 5 d^{2}}} = \frac{a}{b}

(iii)

\frac{a^{2} + a b + b^{2}}{a^{2} - a b + b^{2}} = \frac{c^{2} + c d + d^{2}}{c^{2} - c d + d^{2}}

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\frac{a}{b} = \frac{c}{d} = \frac{e}{f}

prove that :

(b^{2} + d^{2} + f^{2}) + (a^{2} + c^{2} + e^{2}) = (a b + c d + e f)^{2}

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