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.

According to question,

$\Rightarrow \dfrac{x}{y} = \dfrac{(x + z)^2}{(y + z)^2} \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{x^2 + z^2 + 2xz}{y^2 + z^2 + 2yz} \\[1em] \Rightarrow x(y^2 + z^2 + 2yz) = y(x^2 + z^2 + 2xz) \\[1em] \Rightarrow xy^2 + xz^2 + 2xyz = yx^2 + yz^2 + 2xyz \\[1em] \Rightarrow xz^2 - yz^2 = 2xyz - 2xyz + yx^2- xy^2 \\[1em] \Rightarrow z^2(x - y) = xy(x - y) \\[1em] \Rightarrow z^2 = xy.$

Since, z² = xy hence, proved that z is mean proportional between x and y.