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Xét \(\left(a^2+b^2\right).C-\left(b^2+c^2\right).a=a^2c+b^2a\)=\(b^2a-c^2a=a^2c+ac.c-ac.a=0\)
(thay \(b^2=ac\))
\(\Rightarrow\left(a^2+b^2\right).c=\left(b^2+c^2\right).a\Rightarrow\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{a}{c}\)
\(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\)
Áp dụng t/c dtsbn:
\(\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{a-b}{c-d}=\dfrac{a+b}{c+d}\)
\(\Rightarrow\dfrac{a-b}{c-d}=\dfrac{a+b}{c+d}\Rightarrow\dfrac{a-b}{a+b}=\dfrac{c-d}{c+d}\)
\(\dfrac{a}{b}=\dfrac{c}{d}=>\dfrac{a}{b}+1=\dfrac{c}{d}+1=>\dfrac{a+b}{b}=\dfrac{c+d}{d}\)
\(\dfrac{a}{b}=\dfrac{c}{d}=>\dfrac{a}{b}-1=\dfrac{c}{d}-1=>\dfrac{a-b}{b}=\dfrac{c-d}{d}\)
\(\dfrac{a}{b}=\dfrac{c}{d}=>ad=cb=>ad+ac=cb+ac\)
\(=>a\left(c+d\right)=c\left(a+b\right)=>\dfrac{a}{c}=\dfrac{a+b}{c+d}=>\dfrac{a}{a+b}=\dfrac{c}{c+d}\)
\(a,\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{a+c}{b+d}\\ b,\dfrac{a}{b}=\dfrac{c}{d}\Leftrightarrow\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\\ \Leftrightarrow\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\)
2 . a ) nếu \(\dfrac{a}{b}< \dfrac{c}{d}\)thì a.d < b.c
\(\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd}\Rightarrow\dfrac{a}{bd}< \dfrac{c}{bd}\Rightarrow a< c\)
vì a<c => a.d < b.c
=> đcpm
b) ko ghi lại đề
vì a.d<c.d => \(\dfrac{a}{bd}< \dfrac{c}{bd}\Rightarrow\dfrac{a}{b}< \dfrac{c}{d}\)( bn suy luận ngược với a nhé )
Có: \(a+b+c=1\Leftrightarrow\left(a+b+c\right)^2=1\)
Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:
\(\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}=\dfrac{x+y+z}{a+b+c}\)
\(\Rightarrow\dfrac{x^2}{a^2}=\dfrac{y^2}{b^2}=\dfrac{z^2}{c^2}=\dfrac{\left(x+y+z\right)^2}{\left(a+b+c\right)^2}=\dfrac{x^2+y^2+z^2}{a^2+b^2+c^2}\)
\(\Rightarrow\left(x+y+z\right)^2=x^2+y^2+z^2\) (do \(\left(a+b+c\right)^2=a^2+b^2+c^2=1\))
Có \(\dfrac{a}{b}=\dfrac{c}{d}=>ad=bc\) => a2 = ad => a=d
Xét \(\dfrac{a+b}{a-b}=\dfrac{c+a}{c-a}\)
<=> (a+b)(c-a) = (a-b)(c+a)
<=> (a+b)(c-d) = (a-b)(c+d)
<=> ac - ad + bc - bd = ac + ad -bc -bd
<=> 2bc = 2ad (luôn đúng) => đpcm
cảm ơn