Cho a, b, c khác nhau đôi một, chứng minh rằng:
\(\frac{{b - c}}{{\left( {a - b} \right)\left( {a - c} \right)}} + \frac{{c - a}}{{\left( {b - c} \right)\left( {b - a} \right)}} + \frac{{a - b}}{{\left( {c - a} \right)\left( {c - b} \right)}} = \frac{2}{{a - b}} + \frac{2}{{b - c}} + \frac{2}{{c - a}}\).
VT = \(\frac{{b - c}}{{\left( {a - b} \right)\left( {a - c} \right)}} + \frac{{c - a}}{{\left( {b - c} \right)\left( {b - a} \right)}} + \frac{{a - b}}{{\left( {c - a} \right)\left( {c - b} \right)}}\)
VT =\(\frac{{b - a + a - c}}{{\left( {a - b} \right)\left( {a - c} \right)}} + \frac{{c - b + b - a}}{{\left( {b - c} \right)\left( {b - a} \right)}} + \frac{{a - c + c - b}}{{\left( {c - a} \right)\left( {c - b} \right)}}\)
VT =\(\frac{{ - 1}}{{a - c}} + \frac{1}{{a - b}} + \frac{{ - 1}}{{b - a}} + \frac{1}{{b - c}} + \frac{{ - 1}}{{c - b}} + \frac{1}{{c - a}}\)
VT =\(\frac{2}{{a - b}} + \frac{2}{{b - c}} + \frac{2}{{c - a}}\)
Suy ra: \(\frac{{b - c}}{{\left( {a - b} \right)\left( {a - c} \right)}} + \frac{{c - a}}{{\left( {b - c} \right)\left( {b - a} \right)}} + \frac{{a - b}}{{\left( {c - a} \right)\left( {c - b} \right)}} = \frac{2}{{a - b}} + \frac{2}{{b - c}} + \frac{2}{{c - a}}\).