If ` a : b = b : c`, then prove that
`(abc(a+b+c)^(3))/((ab+bc+ca)^(3)) = 1`

Explanation

Given that ` a : b = b : c`
` rArr a/b = b/c = k " (let)", [k ne 0]`
` :. A = bk = ck.k = ck^(2) " and " b = ck`.
LHS ` = (abc (a + b + c)^(3))/((ab+bc + ca)^(3) ) = (ck^(2).ck.c(ck^(2)+ck+c)^(3))/((ck^(2).ck+ck.c + c.ck^(2))^(3))`
`= (c^(3)k^(3){c(k^(2)+k+1)}^(3))/((c^(2)k^(3)+c^(2)k+c^(2)k^(2))^(3))=(c^(6)k^(3)(k^(2)+k+1)^(3))/(c^(6)k^(3)(k^(2)+k+1)^(3)) = 1` RHS = 1.
` :. ` LHS = RHS .