Proof:


We will prove the given equation using algebraic manipulation.

Step 1:


Expand the left side of the equation using the identity (a+b+c)^3 = a^3+b^3+c^3+3ab(a+b)+3bc(b+c)+3ca(c+a)+6abc.

(a+b+c)^3 - a^3 - b^3 - c^3
= 3(ab(a+b) + bc(b+c) + ca(c+a) + 2abc)

Step 2:


Factor out 3abc from the right side of the equation.

3(ab(a+b) + bc(b+c) + ca(c+a) + 2abc)
= 3abc(a+b+c) + 3abc(ab/c + bc/a + ca/b + 2)

Step 3:


Substitute (ab/c + bc/a + ca/b) with the identity [(a/b)^2 + (b/c)^2 + (c/a)^2 - (a/c)^2 - (b/a)^2 - (c/b)^2].

3abc(a+b+c) + 3abc[(a/b)^2 + (b/c)^2 + (c/a)^2 - (a/c)^2 - (b/a)^2 - (c/b)^2 + 2]
= 3abc(a+b+c) + 3abc[(a/b)^2 + (b/c)^2 + (c/a)^2 + 2 - (a/c)^2 - (b/a)^2 - (c/b)^2]

Step 4:


Simplify the right side of the equation by using the identity (a/b)^2 + (b/c)^2 + (c/a)^2 + 2 = (a/b + b/c + c/a)^2.

3abc(a+b+c) + 3abc[(a/b + b/c + c/a)^2 - (a/c)^2 - (b/a)^2 - (c/b)^2]
= 3abc(a+b+c) + 3abc[(a/b + b/c + c/a + a/c + b/a + c/b)(a/b + b/c + c/a - a/c - b/a - c/b)]

Step 5:


Simplify the right side of the equation by using the identity (a/b + b/c + c/a)^2 - (a/c)^2 - (b/a)^2 - (c/b)^2 = 2(a/b)(b/c)(c/a).

3abc(a+b+c) + 3abc[(a/b + b/c + c/a + a/c + b/a + c/b)(a/b + b/c + c/a - a/c - b/a - c/b)]
= 3abc(a+b+c) + 6abc(a/b)(b/c)(c/a)

Step 6:


Factor out 3abc from both sides of the equation.

(a+b+c)^3 - a^3 - b^3 - c^3 = 3(a/b)(b/c)(c/a)

Therefore, the given equation is proved.