2012(x + y) = 2013(y + z) = 2014 (z + x)
\(=\frac{x+y}{\frac{1}{2012}}=\frac{y+z}{\frac{1}{2013}}=\frac{z+x}{\frac{1}{2014}}\)
Áp dụng tính chất của dãy tỉ số = nhau ta có:
\(\frac{x+y}{\frac{1}{2012}}=\frac{y+z}{\frac{1}{2013}}=\frac{z+x}{\frac{1}{2014}}=\frac{\left(z+x\right)-\left(y+z\right)}{\frac{1}{2014}-\frac{1}{2013}}=\frac{\left(y+z\right)-\left(x+y\right)}{\frac{1}{2013}-\frac{1}{2012}}\)
\(=\frac{x-y}{\frac{-1}{2013.2014}}=\frac{z-x}{\frac{-1}{2012.2013}}\)
= (x - y).(-2013.2014) = (z - x).(-2012.2013)
=> (x - y).(-2013.2014).\(\frac{-1}{2013.2014.1006}\) = (z - x).(-2012.2013).\(\frac{-1}{2013.2014.1006}\)
\(\Rightarrow\frac{x-y}{1006}=\frac{z-x}{1007}\left(đpcm\right)\)
