Using properties of determinants prove that:

[R₁’ = R₁ + R₂ + R₃]

[R₁’ = R₁/(a + b + c)]

= (a + b + c)[2(b - c)c - b(c - a) + (c + a)(c - a) - (a + b)(b - c)] [expansion by first row]

= (a + b + c)(2bc - 2c² - bc + ab + c² - a² - ab - b² + ac + bc

= (a + b + c)(ab + bc + ac - a² - b² - c²)

= 3abc - a³ - b³ - c³