Toán Lớp 7: Cho x, y, z khác 0, thỏa mãn $\frac{x-y}{x+y}$ = $\frac{z-x}{z+x}$ . Chứng minh rằng x ²=yz. Help me;-;
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Toán Lớp 7: Cho x, y, z khác 0, thỏa mãn $\frac{x-y}{x+y}$ = $\frac{z-x}{z+x}$ . Chứng minh rằng x ²=yz. Help me;-;
Comments ( 1 )
\text{Giả sử}:{x^2} = y.z\\
\Leftrightarrow \dfrac{x}{y} = \dfrac{z}{x} = k \Leftrightarrow \left\{ \begin{array}{l}
x = k.y\\
z = k.x
\end{array} \right.\\
+ \dfrac{{x – y}}{{x + y}} = \dfrac{{k.y – y}}{{k.y + y}} = \dfrac{{\left( {k – 1} \right).y}}{{\left( {k + 1} \right).y}} = \dfrac{{k – 1}}{{k + 1}}\\
+ \dfrac{{z – x}}{{z + x}} = \dfrac{{k.x – x}}{{k.x + x}} = \dfrac{{\left( {k – 1} \right).x}}{{\left( {k + 1} \right).x}} = \dfrac{{k – 1}}{{k + 1}}\\
\Leftrightarrow \dfrac{{x – y}}{{x + y}} = \dfrac{{z – x}}{{z + x}}
\end{array}$