Question

Nếu \(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\)thì x;y;z tương ứng tỉ lệ với a;b;c

Answer 1

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

\(\Leftrightarrow\dfrac{abz-acy}{a^2}=\dfrac{bcx-caz}{b^2}=\dfrac{cay-cbx}{c^2}\)

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

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

Do đó :

\(abz=acy\Leftrightarrow bz=cy\Leftrightarrow\dfrac{z}{c}=\dfrac{y}{b}\left(1\right)\)

\(bcx=baz\Leftrightarrow cx=az\Leftrightarrow\dfrac{x}{a}=\dfrac{z}{c}\left(2\right)\)

Từ \(\left(1\right)+\left(2\right)\Leftrightarrow\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}\rightarrowđpcm\)