(a+b+c+d)(a+d-b-c)=(a-b+c-d)(a+b-c-d)

=>(a+d)^2-(b+c)^2=(a-d)^2-(b-c)^2

=>(a+d)^2-(a-d)^2=(b+c)^2-(b-c)^2

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

=>4ad=4bc

=>ad=bc

=>a/c=b/d

a,

b,  a/b < c/d => ad < cb
=>ad +ab < bc+ab
=> a(d+b) < b(a+c)
=> a/b < a+c/d+b (1)
* a/b < c/d => ad<cb
=> ad + cd < cb +cd
=> d(a+c) < c(b+d) 
=> c/d > a+c/b+d (2)
Từ (1) và (2) => a/b < a+c/b+d < c/d

Vì \(b,d>0\)nên \(bd>0\)

Ta có:  \(\frac{a}{b}< \frac{c}{d}\)

\(\Leftrightarrow\frac{ad}{bd}< \frac{bc}{bd}\)

\(\Leftrightarrow ad< bc\)vì \(bd>0\)

Ta có: \(\left(a+b+c-d\right)\left(a-b-c-d\right)=\left(a+b-c+d\right)\left(a-b+c+d\right)\)

\(\Rightarrow\frac{a+b+c-d}{a+b-c+d}=\frac{a-b+c+d}{a-b-c-d}\Leftrightarrow\frac{\left(a+b\right)+\left(c-d\right)}{\left(a+b\right)-\left(c-d\right)}=\frac{\left(a-b\right)+\left(c+d\right)}{\left(a-b\right)-\left(c+d\right)}.\)

Đặt \(A=a+b;B=c-d;C=a-b;D=c+d.\)Ta được:

\(\frac{A+B}{A-B}=\frac{C+D}{C-D}\Rightarrow\frac{A}{B}=\frac{C}{D}\Leftrightarrow\frac{a+b}{c-d}=\frac{a-b}{c+d}\Rightarrow\frac{a+b}{a-b}=\frac{c-d}{c+d}\)

Vậy ta được:

\(\left(a+b+c-d\right)\left(a-b-c-d\right)=\left(a+b-c+d\right)\left(a-b+c+d\right)\)

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

đặt k=a/b=c/d => a=bk;c=dk

=> \(\frac{a+b}{b}=\frac{b+bk}{b}=\frac{b\left(1+k\right)}{b}=1+k\)

=>\(\frac{c+d}{d}=\frac{dk+d}{d}=\frac{d\left(k+1\right)}{d}=k+1\)

=>nếu a/b=c/d thì a+b/b = c+d/d

Ta có: \(\frac{a}{b}< \frac{c}{d}\Leftrightarrow ad< bc\Leftrightarrow ad+ab< bc+ab\Leftrightarrow a\left(d+b\right)< b\left(c+a\right)\Leftrightarrow\frac{a}{b}< \frac{a+c}{b+d}\)(1)

\(\frac{a}{b}< \frac{c}{d}\Leftrightarrow bc>ad\Leftrightarrow bc+cd>ad+cd\Leftrightarrow c\left(b+d\right)>d\left(a+c\right)\Leftrightarrow\frac{c}{d}>\frac{a+c}{b+d}\)(2)

Từ (1) và (2) suy ra điều phải chứng minh