(x-b)(x-c)/(a-b)(a-c) +
(x-c)(x-a)/(b-c)(b-a) +
(x-a)(x-b)/(c-a)(c-b)
= (x-b)(x-c)(c-b)/(a-b)(b-c)(c-a) +
(x-c)(x-a)(a-c)/(a-b)(b-c)(c-a) +
(x-a)(x-b)(b-a)/(a-b)(b-c)(c-a)
=[ (xx-bx-ax+ab)(b-a) +
(a->b b->c) +
(a->c b->a)
]/(a-b)(b-c)(c-a)
=[ ( (b-a)xx-(bb-aa)x+ab(b-a) )+
(a->b b->c) +
(a->c b->a)
]/(a-b)(b-c)(c-a)
xx terms and x terms cancel due to cyclic property
=[ ab(b-a)+bc(c-b)+ca(a-c)]/(a-b)(b-c)(c-a)
=(abb-aab+bcc-bbc+caa-cca)/(ab-ac-bb+bc)(c-a)
=(abb-aab+bcc-bbc+caa-cca)/(abc-aab-acc+aac-bbc+bba+bcc-abc)
=(abb-aab+bcc-bbc+caa-cca)/(-aab-acc+aac-bbc+bba+bcc)
=1