Question

Cho \(\dfrac{yc-bz}{x}=\dfrac{za-xc}{y}=\dfrac{xb-ya}{z}\). CMR \(\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}\)

Answer 1

\(\dfrac{yc-bz}{x}=\dfrac{za-xc}{y}=\dfrac{xb-ya}{z}\)

\(\Rightarrow\dfrac{xyc-xbz}{x^2}=\dfrac{yza-xyc}{y^2}=\dfrac{xbz-yza}{z^2}\)

Áp dụng tính chất dãy tỉ số bằng nhau:

\(\dfrac{xyc-xbz}{x^2}=\dfrac{yza-xyc}{y^2}=\dfrac{xbz-yza}{z^2}\)

\(=\dfrac{xyc-xbz+yza-xyc+xbz-yza}{x^2+y^2+z^2}=0\)

\(\Rightarrow\left\{{}\begin{matrix}yc=bz\\za=xc\\xb=ya\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{y}{b}=\dfrac{z}{c}\\\dfrac{x}{a}=\dfrac{z}{c}\\\dfrac{x}{a}=\dfrac{y}{b}\end{matrix}\right.\Leftrightarrow\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}\left(đpcm\right)\)