P=9

Do \(\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}\\ \)

=> \(\frac{b+c}{a}=\frac{c+a}{b}=\frac{a+b}{c}\)

Áp dụng tính chất dãy tỉ số bằng nhau, ta có:

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

=> \(\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}=2+2+2=6\)

Giúp mk với mk đang cần gấp lắm

\(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\)

=> \(\frac{2}{c}=\frac{1}{a}+\frac{1}{b}\)

=> \(\frac{2}{c}=\frac{a+b}{ab}\)

=> 2ab = ac + bc

=> ac + bc - 2ab = 0

=> (ac - ab) + (bc - ab) = 0

=> a(c - b) + b(c - a) = 0

=> a(c - b) = -b(c - a)

=> a(c - b) = b(a - c)

=> \(\frac{a}{b}=\frac{a-c}{c-b}\) (đpcm)

Ta có:

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

\(\Rightarrow a=b=c\)

\(\Rightarrow M=\frac{a^3.b^2.c^{1930}}{b^{1935}}=\frac{b^{1935}}{b^{1935}}=1\)

Áp dụng tỉ dãy số bằng nhau:

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

Khi đó: \(\frac{a^3.b^2.c^{1930}}{b^{1935}}\Leftrightarrow\frac{b^{1935}}{b^{1935}}=b^{1935}:b^{1935}=1\)

\(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\)

=>\(\frac{1}{c}=\frac{a+b}{2ab}\)

=> 2ab = c(a+b)

=> ab+ab = ac+bc

=> ab - bc = ac - ab

=> b(a-c) = a(c-b)

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

Có: \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\)

Đặt \(\frac{a}{c}=\frac{b}{d}=k\left(1\right)\\ \Rightarrow\left\{{}\begin{matrix}a=ck\\b=dk\end{matrix}\right.\)

\(\frac{a-b}{c-d}=\frac{ck-dk}{c-d}=\frac{k\left(c-d\right)}{c-d}=k\left(2\right)\)

(1)(2) \(\Rightarrow\frac{a}{c}=\frac{a-b}{c-d}\)

Cảm ơn bạn nhiều nha

ta có:\(\frac{a}{b}=\frac{c}{d}\) \(\Rightarrow\frac{b}{a}=\frac{c}{d}\)

                               \(\Rightarrow1-\frac{b}{a}=1-\frac{c}{d}\)

                                \(\Rightarrow\frac{a}{a}-\frac{b}{a}=\frac{c}{c}-\frac{d}{c}\)

                                \(\Rightarrow\frac{a-b}{a}=\frac{c-d}{c}\)

                            hay: \(\frac{a}{a-b}=\frac{c}{c-d}\)(đpcm)

Cách 1 : \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{a-b}{c-d}\Rightarrow\frac{a}{a-b}=\frac{c}{c-d}\)

Cách 2 : \(\frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc\Rightarrow ac-ad=ac-bc\)

\(\Rightarrow a(c-d)=c(a-b)\Rightarrow\frac{a}{a-b}=\frac{c}{c-d}\)

Cách 3 : Đặt \(\frac{a}{b}=\frac{c}{d}=m\Rightarrow a=mb,c=md\)

Ta có : \(\frac{a}{a-b}=\frac{mb}{mb-b}=\frac{mb}{b(m-1)}=\frac{m}{m-1}\)

\(\frac{c}{c-d}=\frac{md}{md-d}=\frac{md}{d(m-1)}=\frac{m}{m-1}\)

Do đó : \(\frac{a}{a-b}=\frac{c}{c-d}\)

Cách 4 : \(\frac{a}{a-b}=\frac{c}{c-d}\Rightarrow a(c-d)=c(a-b)\)

\(\Rightarrow ac-ad=ac-bc\Rightarrow ad=bc\Leftrightarrow\frac{a}{b}=\frac{c}{d}\) đẳng thức đúng

Do đó , ta có : \(\frac{a}{a-b}=\frac{c}{c-d}\)là đẳng thức đúng.