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VP = \(\dfrac{\left(a-b\right)^2}{\left(a+c\right)\left(b+c\right)}+\dfrac{\left(b-c\right)^2}{\left(b+a\right)\left(c+a\right)}+\dfrac{\left(c-a\right)^2}{\left(c+b\right)\left(a+b\right)}\)
\(=\left(a-b\right).\dfrac{\left(a+c\right)-\left(b+c\right)}{\left(a+c\right)\left(b+c\right)}+\left(b-c\right).\dfrac{\left(b+a\right)-\left(c+a\right)}{\left(b+a\right)\left(c+a\right)}+\left(c-b\right).\dfrac{\left(c+b\right)-\left(a+b\right)}{\left(c+b\right)\left(a+b\right)}\)
\(=\left(a-b\right).\left(\dfrac{1}{b+c}-\dfrac{1}{a+c}\right)+\left(b-c\right)\left(\dfrac{1}{c+a}-\dfrac{1}{b+a}\right)+\left(c-a\right).\left(\dfrac{1}{a+b}-\dfrac{1}{c+b}\right)\)
\(=\left(a-b\right).\dfrac{1}{b+c}-\left(a-b\right).\dfrac{1}{a+c}+\left(b-c\right).\dfrac{1}{c+a}-\left(b-c\right).\dfrac{1}{b+a}+\left(c-a\right).\dfrac{1}{a+b}-\left(c-a\right).\dfrac{1}{c+b}\)
\(=\left(2a-b-c\right).\dfrac{1}{b+c}+\left(2b-c-a\right).\dfrac{1}{c+a}+\left(2c-a-b\right).\dfrac{1}{a+b}\)
\(=\dfrac{2a}{b+c}-\left(b+c\right).\dfrac{1}{b+c}+\dfrac{2b}{c+a}-\left(c+a\right).\dfrac{1}{c+a}+\dfrac{2c}{a+b}-\left(a+b\right).\dfrac{1}{a+b}\)
\(=2\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)-3\left(đpcm\right)\)
\(VT=\dfrac{2a^3-a^2b-a^2c-ab^2-ac^2+2b^3-b^2c-bc^2+2c^3}{(a+b)(b+c)(c+a)} \)
\(\\=\dfrac{a^3+a^2b-2a^2b-2ab^2+ab^2+b^3+b^3+b^2c-2b^2c-2bc^2+bc^2+c^3+c^3+c^2a-2c^a+2ca^2-ca^2+a^3}{(a+b)(b+c)(c+a)}\)
\(\\=\dfrac{(a-b)^2(a+b)+(b-c)^2(b+c)+(c-a)^2(c+a)}{(a+b)(b+c)(c+a)}\)
\(\\\Rightarrow VT=\dfrac{(a-b)^2}{(c+a)(b+c)}+\dfrac{(b-c)^2}{(c+a)(a+b)}+\dfrac{(c-a)^2}{(a+b)(b+c)}=VP\)
\((\dfrac{1}{\left(b-c\right)\left(a^2+ac-b^2-bc\right)}+\dfrac{1}{\left(c-a\right)\left(b^2+ba-c^2-ca\right)}+\dfrac{1}{\left(a-b\right)\left(c^2+cb-a^2-ab\right)}=0 \)
\(\Leftrightarrow\dfrac{1}{\left(b-c\right)\left[\left(a-b\right)\left(a+b\right)+c\left(a-b\right)\right]}+\dfrac{1}{\left(c-a\right)\left[\left(b-c\right)\left(b+c\right)+a\left(b-c\right)\right]}+\dfrac{1}{\left(a-b\right)\left[\left(c-a\right)\left(c+a\right)+b\left(c-a\right)\right]}=0\)
\(\Leftrightarrow\dfrac{1}{\left(b-c\right)\left(a-b\right)\left(a+b+c\right)}+\dfrac{1}{\left(c-a\right)\left(b-c\right)\left(a+b+c\right)}+\dfrac{1}{\left(a-b\right)\left(c-a\right)\left(a+b+c\right)}=0\)
\(\Leftrightarrow\dfrac{c-a+a-b+b-c}{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)}=0\)
\(\Leftrightarrow\dfrac{0}{\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)}=0\)(t/m)
Suy ra ta được Đt cần chứng minh.
Chúc bạn học tốt với hoc24 nha
Lời giải:
Ta có:
\(\frac{1}{(b-c)(a^2+ac-b^2-bc)}+\frac{1}{(c-a)(b^2+bc-c^2-ca)}+\frac{1}{(a-b)(c^2+cb-a^2-ab)}\)
\(=\frac{1}{(b-c)[(a^2-b^2)+(ac-bc)]}+\frac{1}{(c-a)[(b^2-c^2)+(ba-ca)]}+\frac{1}{(a-b)[(c^2-a^2)+(cb-ab)]}\)
\(=\frac{1}{(b-c)[(a-b)(a+b)+c(a-b)]}+\frac{1}{(c-a)[(b-c)(b+c)+a(b-c)]}+\frac{1}{(a-b)[(c-a)(c+a)+b(c-a)]}\)
\(=\frac{1}{(b-c)(a-b)(a+b+c)}+\frac{1}{(c-a)(b-c)(b+c+a)}+\frac{1}{(a-b)(c-a)(c+a+b)}\)
\(=\frac{(c-a)+(a-b)+(b-c)}{(a-b)(b-c)(c-a)(a+b+c)}=\frac{0}{(a-b)(b-c)(c-a)(a+b+c)}=0\)
Ta có đpcm.
Từ \(\dfrac{a-\left(c-b\right)}{b-c}+\dfrac{b-\left(a-c\right)}{c-a}+\dfrac{c-\left(b-a\right)}{a-b}=3\)
\(=>\dfrac{a}{b-c}+1+\dfrac{b}{c-a}+1+\dfrac{c}{a-b}+1=3\)
\(=>\dfrac{a}{b-c}-\dfrac{b}{a-c}-\dfrac{c}{b-a}=0\)
\(=>\dfrac{a}{b-c}=\dfrac{b}{a-c}+\dfrac{c}{b-a}=\dfrac{b^2-ab+ac-c^2}{\left(c-a\right)\left(a-b\right)}\)
Nhân cả 2 vế với \(\dfrac{1}{b-c}\) ta được
\(\dfrac{a}{\left(b-c\right)^2}=\dfrac{b^2-ab+ac-c^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\left(1\right)\)
Tương tự ta có:
\(\dfrac{b}{\left(c-a\right)^2}=\dfrac{c^2-bc+bc-a^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\left(2\right)\)
\(\dfrac{c}{\left(a-b\right)^2}=\dfrac{a^2-ca+cb-c^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\left(3\right)\)
Cộng theo vế (1);(2);(3) ta có ĐPCM
CHÚC BẠN HỌC TỐT.........
1)\(\dfrac{c-b}{\left(a-b\right)\left(c-b\right)\left(a-c\right)}+\dfrac{a-c}{\left(b-a\right)\left(b-c\right)\left(a-c\right)}+\dfrac{b-a}{\left(b-a\right)\left(c-b\right)\left(c-a\right)}=\dfrac{c-b+a-c+b-c}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=0\)
\(A=\dfrac{\left(a-b\right)^2}{ab}+\dfrac{\left(b-c\right)^2}{bc}+\dfrac{\left(c-a\right)^2}{ca}\)
\(B=\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)
..................................
\(A=\dfrac{a^2+b^2-2ab}{ab}+\dfrac{b^2-2ab+c^2}{bc}+c^2+a^2-\dfrac{2ca}{ca}\)
\(A=\left(\dfrac{a}{b}+\dfrac{b}{a}-2\right)+\left(\dfrac{b}{c}+\dfrac{c}{b}-2\right)+\left(\dfrac{c}{a}+\dfrac{a}{c}-2\right)=\dfrac{\left(b+c\right)}{a}+\dfrac{a+c}{b}+\dfrac{a+b}{c}-6\)
\(A=\left[\dfrac{\left(b+c\right)}{a}+1\right]+\left[\dfrac{\left(a+c\right)}{b}+1\right]+\left[\dfrac{\left(a+b\right)}{c}+1\right]-9\)
\(A=\dfrac{\left(a+b+c\right)}{a}+\dfrac{\left(a+b+c\right)}{b}+\left[\dfrac{\left(a+b+c\right)}{c}\right]-9\)
\(A=\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)-9\)
Ket luan
\(A\ne B\) => đề sai--> hoặc mình công trừ sai
Ta có \(\dfrac{2}{a-b}\)+\(\dfrac{2}{b-c}\)+\(\dfrac{2}{c-a}\)
= (\(\dfrac{1}{a-b}\)+\(\dfrac{1}{c-a}\))+(\(\dfrac{1}{b-c}\)+\(\dfrac{1}{a-b}\))+(\(\dfrac{1}{c-a}\)+\(\dfrac{1}{b-c}\))
=(\(\dfrac{1}{a-b}\)- \(\dfrac{1}{a-c}\))+(\(\dfrac{1}{b-c}\)- \(\dfrac{1}{b-a}\))+(\(\dfrac{1}{c-a}\) - \(\dfrac{1}{c-b}\))
=\(\dfrac{\left(a-c\right)-\left(a-b\right)}{\left(a-b\right).\left(a-c\right)}\)+\(\dfrac{\left(b-a\right)-\left(b-c\right)}{\left(b-a\right).\left(b-c\right)}\)+\(\dfrac{\left(c-b\right)-\left(c-a\right)}{\left(c-b\right).\left(c-a\right)}\)
= \(\dfrac{a-c-a+b}{\left(a-b\right).\left(a-c\right)}\)+\(\dfrac{b-a-b+c}{\left(b-a\right).\left(b-c\right)}\)+\(\dfrac{c-b-c+a}{\left(c-b\right).\left(c-a\right)}\)
= \(\dfrac{-c+b}{\left(a-b\right).\left(a-c\right)}\)+ \(\dfrac{-a+c}{\left(b-a\right).\left(b-c\right)}\)+\(\dfrac{-b+a}{\left(c-b\right).\left(c-a\right)}\)
= \(\dfrac{b-c}{\left(a-b\right).\left(a-c\right)}\)+\(\dfrac{c-a}{\left(b-a\right).\left(b-c\right)}\)+\(\dfrac{a-b}{\left(c-b\right).\left(c-a\right)}\)
Chúc bạn học tốt.