Question

Biết  \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\left(a,b,c\ne0\right)\).  Chứng minh: \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)

Answer 1

\(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\) => \(\frac{a.\left(bz-cy\right)}{a^2}=\frac{b.\left(cx-az\right)}{b^2}=\frac{c.\left(ay-bx\right)}{c^2}\)

<=> \(\frac{abz-acy}{a^2}=\frac{bcx-abz}{b^2}=\frac{cay-bcx}{c^2}\). Theo tính chất dãy tỉ số bằng nhau

=> \(\frac{abz-acy}{a^2}=\frac{bcx-abz}{b^2}=\frac{cay-bcx}{c^2}=\frac{abz-acy+bcx-abz+cay-bcx}{a^2+b^2+c^2}=0\)

=> \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\) = 0 

=> \(bz-cy=0\Rightarrow bz=cy\Rightarrow\frac{y}{b}=\frac{z}{c}\) (1)

\(cx-az=0\Rightarrow\frac{x}{a}=\frac{z}{c}\)  (2)

Từ (1)(2) => \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)