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If a, b, c and d are in proportion, then...

If a, b, c and d are in proportion, then show that `(a^(3)+3ab^(2))/(3a^(2)b+b^(3))=(c^(3)+3cd^(2))/(3c^(2)d+d^(3))`

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Given, `(a^(3)+3ab^(2))/(3a^(2)b+b^(3))=(c^(3)+3cd^(2))/(3c^(2)d+d^(3))`
Applying componendo and dividendo,
`implies((a^(3)+3ab^(2))+(3a^(2)b+b^(3)))/((a^(3)+3ab^(2))-(3a^(2)b+b^(3)))`
`=((c^(3)+3cd^(2))+(3c^(2)d+d^(3)))/((c^(3)+3cd^(2))-(3c^(2)d+d^(3)))`
`implies(a^(3)+b^(3)+3ab(a+b))/(a^(3)-b^(3)-3ab(a-b))=(c^(3)+d^(3)+3cd(c+d))/(c^(3)-d^(3)-3cd(c-d))`
`implies((a+b)^(3))/((a-b)^(3))=((c+d)^(3))/(c-d)^(3))implies(a+b)/(a-b)=(c+d)/(c-d)`
Again applying componendo and dividendo,
`implies((a+b)+(a-b))/((a+b)-(a-b))=((c+d)+(c-d))/((c+d)-(c-d))`
`implies(2a)/(2b)=(2c)/(2d)implies(a)/(b)=(c)/(d)`
`thereforea,b,c and d` are in proportion.
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