Question

Cho \(\frac{a-b}{b-c}\)\(=\)\(\frac{c-d}{d-a}\)CMR : a = c hoặc a + c = b + d

Answer 1

TH1 : \(a-b=c-d=0\)

\(\Rightarrow a=b;c=d\)

\(\Rightarrow a+c=b+d\)

TH2 :\(a-b\ne0;c-d\ne0\)

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

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

\(\Rightarrow ad-a^2-bd+ab=bc-bd-c^2+cd\)

\(\Rightarrow ad-a^2+ab=bc-c^2+cd\)

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

Với  \(d-a+b=b-c+d=0\)

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

\(\Rightarrow a=c\)

Với \(d-a+b\ne0;b-c+d\ne0\)

\(\Rightarrow a=c\)

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