Ta có : \(\frac{b-c}{\left(a-b\right)\left(a-c\right)}=\frac{\left(a-c\right)-\left(a-b\right)}{\left(a-b\right)\left(a-c\right)}=\frac{a-c}{\left(a-b\right)\left(a-c\right)}-\frac{a-b}{\left(a-b\right)\left(a-c\right)}\)

\(=\frac{1}{a-b}-\frac{1}{a-c}=\frac{1}{a-b}+\frac{1}{c-a}\left(1\right)\)

Tương tự ta cũng chứng minh được :

\(\hept{\begin{cases}\frac{c-a}{\left(b-c\right)\left(b-a\right)}=\frac{1}{b-c}+\frac{1}{a-b}\left(2\right)\\\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{1}{c-a}+\frac{1}{b-c}\left(3\right)\end{cases}}\)

Từ (1), (2), (3), suy ra : \(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}\)

\(=\frac{1}{a-b}+\frac{1}{c-a}+\frac{1}{b-c}+\frac{1}{a-b}+\frac{1}{c-a}+\frac{1}{b-c}\)

\(=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\left(đpcm\right)\)

\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}=\frac{c-a+a-b}{\left(a-b\right)\left(c-a\right)}\)=\(\frac{1}{a-b}+\frac{1}{c-a}\)

Tuong tu => DPCM

https://olm.vn/hoi-dap/detail/55826890240.html

\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}=\frac{a-c-\left(a-b\right)}{\left(a-b\right)\left(a-c\right)}=\frac{1}{a-b}+\frac{1}{c-a}\)

\(\frac{c-a}{\left(b-c\right)\left(b-a\right)}=\frac{\left(b-a\right)-\left(b-c\right)}{\left(b-c\right)\left(b-a\right)}=\frac{1}{b-c}+\frac{1}{a-b}\)

\(\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{\left(c-b\right)-\left(c-a\right)}{\left(c-a\right)\left(c-b\right)}=\frac{1}{c-a}+\frac{1}{b-c}\)

Cộng theo vế 3 đẳng thức trên ta có đpcm.

Ta có: 

\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}=\frac{b-a+a-c}{\left(a-b\right)\left(a-c\right)}=\frac{b-a}{\left(a-b\right)\left(a-c\right)}+\frac{a-c}{\left(a-b\right)\left(a-c\right)}=\frac{1}{c-a}+\frac{1}{a-b}\)

Tương tự:

 \(\frac{c-a}{\left(b-c\right)\left(b-a\right)}=\frac{c-b+b-a}{\left(b-c\right)\left(b-a\right)}=\frac{c-b}{\left(b-c\right)\left(b-a\right)}+\frac{b-a}{\left(b-c\right)\left(b-a\right)}=\frac{1}{a-b}+\frac{1}{b-c}\)

Và: \(\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{a-c+c-b}{\left(c-a\right)\left(c-b\right)}=\frac{a-c}{\left(c-a\right)\left(c-b\right)}+\frac{c-b}{\left(c-a\right)\left(c-b\right)}=\frac{1}{b-c}+\frac{1}{c-a}\)

=> \(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{1}{c-a}+\frac{1}{a-b}+\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{b-c}+\frac{1}{c-a}\)

=> \(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)

=> đpcm

bo ko biet

Câu hỏi của Hoàng Minh Nguyễn - Toán lớp 9 - Học toán với OnlineMath

Đặt \(A=\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}\)

\(A=\frac{\left(c-b\right)\left(b-c\right)+\left(c-a\right)\left(a-c\right)+\left(a-b\right)\left(b-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(A=\frac{-b^2+2bc-c^2-a^2+2ac-c^2-a^2+2ab-b^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\Rightarrow\frac{A}{2}=\frac{ab+bc+ca-a^2-b^2-c^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

Đặt \(B=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)

\(\frac{B}{2}=\frac{\left(b-c\right)\left(c-a\right)+\left(a-b\right)\left(c-a\right)+\left(a-b\right)\left(b-c\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\frac{B}{2}=\frac{bc-ab-c^2+ac+ac-a^2-bc+ab+ab-ac-b^2+bc}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\frac{B}{2}=\frac{ab+bc+ca-a^2-b^2-c^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\Rightarrow A=B\left(đpcm\right)\)