Three terms are
$G_{1}=\left(\frac{y}{z}\right)^{\log y z} G_{2}=\left(\frac{z}{x}\right)^{\log z x} G_{3}=\left(\frac{x}{y}\right)^{\log x y}$
Geometric mean of three terms is
$m=\sqrt[3]{G_{1} G_{2} G_{3}}$ ...(i)
$\begin{aligned} G_{1} G_{2} G_{3}=&\left(\frac{y}{z}\right)^{\log y z} \cdot\left(\frac{z}{x}\right)^{\log z x} \cdot\left(\frac{x}{y}\right)^{\log x y} \\ &=\frac{y^{\log y} \cdot y^{\log z}}{z^{\log y} \cdot z^{\log z}} \times \frac{z^{\log z} \cdot z^{\log x}}{x^{\log z} \cdot x^{\log x}} \times \frac{x^{\log x} \cdot x^{\log y}}{y^{\log x} \cdot y^{\log y}} \\ &=\left(\frac{y}{x}\right)^{\log z} \cdot\left(\frac{z}{y}\right)^{\log x} \cdot\left(\frac{x}{z}\right)^{\log y} \end{aligned}$
Taking log both sides $\begin{aligned} \log G_{1} G_{2} G_{3}=& \log \left[\left(\frac{y}{x}\right)^{\log z}\right]+\log \left[\left(\frac{z}{y}\right)^{\log x}\right] \\ &+\log \left[\left(\frac{x}{z}\right)^{\log y}\right] \\=& \log z \log y-\log z \log x+\log x \log z \\ &-\log x \log y+\log y \log x-\log y \log z \end{aligned}$
$\log G_{1} G_{2} G_{3}=0$
$G_{1} G_{2} G_{3}=\mathrm{e}^{0}=1$
Hence $m=\sqrt[3]{G_{1} G_{2} G_{3}}=(1)^{\frac{1}{3}}$
$m=1$