Question

[Hoang Anh Dao]

Cho day ti so : \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c};\)

CMR: \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)

Answer 1

[๖Fly༉Donutღღ]

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

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

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

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

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