If ` x : a = y : b = z : c `, then prove that
`(a^(2) + b^(2) + c^(2))(x^(2) + y^(2) + z^(2)) = (ax + by + cz)^(2)`

Explanation

Given that ` x : a = y : b = z : c`
or, ` x/a = y/b = z/c = k (k ne 0) `(let)
` :. X = ak, y = bk , z = ck`.
LHS `(a^(2)+b^(2)+c^(2))(x^(2)+y^(2)+z^(2))`
` = (a^(2) + b^(2) + c^(2)){(ak)^(2)+(bk)^(2)+(ck)^(2)}`
` = (a^(2) + b^(2) + c^(2))(a^(2)k^(2)+b^(2)k^(2)+c^(2)k^(2))`
` = (a^(2) + b^(2) + c^(2))* k^(2) (a^(2)+b^(2)+c^(2)) = k^(2) (a^(2)+b^(2)+c^(2))^(2)`
RHS ` = (ax + by + cz)^(2) = (a.ak+b.bk + c.ck)^(2)`
` = (a^(2)k + b^(2)k + c^(2)k)^(2) = {k(a^(2) + b^(2) +c^(2))}^(2)`
` = k^(2) (a^(2)+b^(2)+c^(2)) `
` :. ` LHS = RHS