Δ = $\left|\begin{array}{ccc}\mathit{a}& \mathit{b}& \mathit{c}\\ \mathit{a}-\mathit{b}& \mathit{b}-\mathit{c}& \mathit{c}-\mathit{a}\\ \mathit{b}+\mathit{c}& \mathit{c}+\mathit{a}& \mathit{a}+\mathit{b}\end{array}\right|$
Applying ${\mathit{C}}_1$ → ${\mathit{C}}_1+{\mathit{C}}_2+{\mathit{C}}_3$
Δ = $\left|\begin{array}{ccc}\mathit{a}+\mathit{b}+\mathit{c}& \mathit{b}& \mathit{c}\\ 0& \mathit{b}-\mathit{c}& \mathit{c}-\mathit{a}\\ 2\left(\mathit{a}+\mathit{b}+\mathit{c}\right)& \mathit{c}+\mathit{a}& \mathit{a}+\mathit{b}\end{array}\right|$
Δ = (a + b + c) $\left|\begin{array}{ccc}1& \mathit{b}& \mathit{c}\\ 0& \mathit{b}-\mathit{c}& \mathit{c}-\mathit{a}\\ 0& \mathit{c}+\mathit{a}-2\mathit{b}& \mathit{a}+\mathit{b}-2\mathit{c}\end{array}\right|$
${\mathit{R}}_3$ → ${\mathit{R}}_3-2{\mathit{R}}_1$
Δ = (a + b + c) $\left|\begin{array}{ccc}1& \mathit{b}& \mathit{c}\\ 0& \mathit{b}-\mathit{c}& \mathit{c}-\mathit{a}\\ 0& \mathit{c}+\mathit{a}-2\mathit{b}& \mathit{a}+\mathit{b}-2\mathit{c}\end{array}\right|$
Expanding along ${\mathit{C}}_1$, we have,
Δ = (a +b +c) ((b –c) (a + b – 2c) – (c – a) (c + a – 2b))
⇒ Δ = (a + b + c) ((ba + ${\mathit{b}}^2$ - 2bc - ca - cb + $2{\mathit{c}}^2$ - (${\mathit{c}}^2$ + ac - 2bc - ac - ${\mathit{a}}^2$ + 2ab))
⇒ Δ = (a + b + c) (${\mathit{a}}^2+{\mathit{b}}^2+{\mathit{c}}^2$ - ca - bc - ab)
⇒ Δ = (a + b + c) (${\mathit{a}}^2+{\mathit{b}}^2+{\mathit{c}}^2$ - ab - bc - ac)
⇒ Δ = ${\mathit{a}}^3+{\mathit{b}}^3+{\mathit{c}}^3$ - 3abc = R.H.S.