$\frac{bc}{(a - b)(a - c)} + \frac{ac}{(b - a)(b - c)} + \frac{ab}{(c - a)(c - b)} = 1$
$\frac{bc}{(a - b)(a - c)} - \frac{ac}{(a - b)(b - c)} + \frac{ab}{(a - c)(b - c)} = 1$
$\frac{bc(b - c) - ac(a - c) + ab(a - b)}{(a - b)(a - c)(b - c)} = 1$
$\frac{b^2c - bc^2 - a^2c + ac^2 + a^2b - ab^2}{(a - b)(a - c)(b - c)} = 1$
$\frac{b^2c - bc^2 - a^2c + ac^2 + a^2b - ab^2}{(a - b)(a - c)(b - c)} = 1$
$\frac{(b^2c - ab^2) - (bc^2 - a^2b) - (a^2c - ac^2)}{(a - b)(a - c)(b - c)} = 1$
$\frac{b^2(c - a) - b(c^2 - a^2) - ac(a - c)}{(a - b)(a - c)(b - c)} = 1$
$\frac{b^2(c - a) - b(c - a)(c + a) + ac(c - a)}{(a - b)(a - c)(b - c)} = 1$
$\frac{(c - a)(b^2 - b(c + a) + ac)}{(a - b)(a - c)(b - c)} = 1$
$\frac{(c - a)(b^2 - bc - ab + ac)}{(a - b)(a - c)(b - c)} = 1$
$\frac{(c - a)((b^2 - bc) - (ab - ac))}{(a - b)(a - c)(b - c)} = 1$
$\frac{(c - a)(b(b - c) - a(b - c))}{(a - b)(a - c)(b - c)} = 1$
$\frac{(c - a)(b - c)(b - a)}{(a - b)(a - c)(b - c)} = 1$
$\frac{(a - c)(b - c)(a - b)}{(a - b)(a - c)(b - c)} = 1$
1 = 1