If a - b - c - d- then according to alternendo property- a - c - b - d-a - d- b - c-b - a - d - c-a - b - b - c - d - d-

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Alternando

If a : b = c ...

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If a : b = c : d, then according to alternendo property

a : c = b : d

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a : d = b : c

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b : a = d : c

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a + b : b = c + d : d

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S = [\begin{matrix} a & b \\ c & d \end{matrix}]

[\begin{matrix} - d & - b \\ - c & a \end{matrix}]

[\begin{matrix} d & - b \\ - c & a \end{matrix}]

[\begin{matrix} d & b \\ c & a \end{matrix}]

[\begin{matrix} d & c \\ b & a \end{matrix}]

If a : b = c : d, then according to alternendo property

a, b, c, d A, B, C, D ϵ R^{*}

\frac{A}{a} = \frac{B}{b} = \frac{C}{c} = \frac{D}{d}

\frac{\sqrt{a A} + \sqrt{b B} + \sqrt{c C} + \sqrt{d D}}{\sqrt{a + b + c + d} \sqrt{A + B + C + D}}

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