If \(y\) is the geometric mean between \(x\) and \(z\), then \(y^2 = xz\) Let the geometric mean between \(x^2 + y^2\) and \(y^2 + z^2\) be \(p\) \(\therefore p^2 = (x^2 + y^2)(y^2 + z^2)\) \(= x^2y^2 + x^2z^2 + y^4 + y^2z^2\) \(= x^2y^2 + y^4 + y^4 + y^2z^2 \quad [\because xz = y^2]\) \(= x^2y^2 + 2y^4 + y^2z^2\) \(= y^2(x^2 + 2y^2 + z^2)\) \(= y^2(x^2 + 2xz + z^2) \quad [\because y^2 = xz]\) \(= y^2(x + z)^2\) \(\therefore p = \sqrt{y^2(x + z)^2}\) \(= y(x + z) = xy + yz\)