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

\(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}=\frac{abx-acy}{a^2}=\frac{bcx-abz}{b^2}=\frac{acy-bcx}{c^2}\)

Áp dụng tính chất dãy tỉ số bằng nhau:

\(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}=\frac{abz-acy}{a^2}=\frac{bcx-abz}{b^2}=\frac{acy-bcx}{c^2}\)

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

=>bz-cy=cx-az=ay-bx=0

bz-cy=0 => bz=cy => \(\frac{b}{y}=\frac{c}{z}\)cx-az=0 => cx=az => \(\frac{c}{z}=\frac{a}{x}\)

=>\(\frac{a}{x}=\frac{b}{y}=\frac{c}{z}\Rightarrow a:b:c=x:y:z\)(đpcm)