[tex] \frac{a}{(a-b)(a-c)} + \frac{b}{(b-c)(b-a)} + \frac{c}{(c-a)(c-b)}=[/tex]
[tex]=\frac{a*(b-c)}{(a-b)(a-c)(b-c)}+ \frac{b*(a-c)}{(b-c)(-1)*(a-b)(a-c)} + \frac{c*(a-b)}{(a-b)(-1)(a-c)*(-1)*(b-c)} [/tex]
[tex]=\frac{a*(b-c)-b*(a-c)+c*(a-b)}{(a-b)(a-c)(b-c)}=\frac{a*b-a*c-b*a+b*c+c*a-c*b}{(a-b)(a-c)(b-c)}[/tex]
[tex]
\frac{0}{(a-b)(a-c)(b-c)}=0[/tex]