Question

cho 3 tỉ số bằng nhau \(\frac{a}{b+c}\)= \(\frac{b}{c+a}\)=\(\frac{c}{a+b}\). CMR ; a=b=c

Answer 1

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

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

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

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

\(c+a=b+\frac{1}{2}a+\frac{1}{2}c\Leftrightarrow\frac{1}{2}a+\frac{1}{2}c=b\left(3\right)\)

Từ (1);(2) và (3)

=> a=b=c (đpcm)