If x and y be unequal and x: y is the duplicate ratio of x + z and y + z, prove that z is mean proportional between x and y. contexto.140 3 years ago Therefore, z is mean proportional between x and y.
![\[\begin{array}{*{35}{l}} x\left( {{y}^{2}}~+\text{ }{{z}^{2}}~+\text{ }2yz \right)\text{ }=\text{ }y\left( {{x}^{2}}~+\text{ }{{z}^{2}}~+\text{ }2xz \right) \\ x{{y}^{2}}~+\text{ }x{{z}^{2}}~+\text{ }2yzx\text{ }=\text{ }{{x}^{2}}y\text{ }+\text{ }{{z}^{2}}y\text{ }+\text{ }2xzy \\ x{{y}^{2}}~+\text{ }x{{z}^{2}}~=\text{ }{{x}^{2}}y\text{ }+\text{ }{{z}^{2}}y \\ xy\left( y\text{ }-\text{ }x \right)\text{ }=\text{ }{{z}^{2}}\left( y\text{ }-\text{ }x \right) \\ xy\text{ }=\text{ }{{z}^{2}} \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-127c71991da47ad7b3133f71aeb0a875_l3.png "Rendered by QuickLaTeX.com")
