If a, b , c, d are positive real numbers such that $a^{2} + b^{2} = c^{2} + d^{2}$ and $a^{2} + d^{2} - a d = b^{2} + c^{2} + b c$ , find the value of $\frac{a b + c d}{a d + b c}$

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Solution

$a^{2} + b^{2} = c^{2} + d^{2} & a^{2} + d^{2} - a d = b^{2} + c^{2} + b c$
$\frac{a b + c d}{a d + b c} = ?$
$b c^{2} + d^{2} - b^{2} + d^{2} - a d = b^{2} + c^{2} + b c$
$2 d^{2} - a d = 2 b^{2} + b c$
$2 d^{2} - 2 b^{2} = a d + b c$
$a d + b c = 2 c d^{2} - b^{2}$
similarly
$a b + c d = 2 (a^{2} - c^{2})$
$\frac{a b + c d}{a d + b c} = \frac{a^{2} - c^{2}}{d^{2} - b^{2}}$

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