Question

Biết rằng: \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)

Hãy chứng minh : \(x:y:z=a:b:c\)

Answer 1

Ta có : 

  \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}=\frac{bxz-cxy}{ax}=\frac{cxy-ayz}{by}\)

             \(=\frac{ayz-bxz}{cz}=\frac{0}{ax+by+cz}=0\)

\(\Leftrightarrow bz=cy\Rightarrow\frac{z}{c}=\frac{y}{b}\)          \(\left(1\right)\)

     \(cx=az\Rightarrow\frac{x}{a}=\frac{z}{c}\)           \(\left(2\right)\)

     \(ay=bx\Rightarrow\frac{y}{b}=\frac{x}{a}\)           \(\left(3\right)\)

Từ \(\left(1\right),\left(2\right),\left(3\right)\Leftrightarrow\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\) hay \(x:y:z=a:b:c\)