Question

Cho

\(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\)
Cm: \(x:y:z=a:b:c\)

Answer 1

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

Nên \(\dfrac{abz-acy}{a^2}=\dfrac{bcx-abz}{b^2}=\dfrac{acy-bcz}{c^2}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

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

Nên \(\left\{{}\begin{matrix}bz=cy\\cx=az\\ay=bx\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{y}{b}=\dfrac{z}{c}\\\dfrac{x}{a}=\dfrac{z}{c}\\\dfrac{x}{a}=\dfrac{y}{b}\end{matrix}\right.\Leftrightarrow\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{c}{z}\)