If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between (a² + b²) and (b² + c²).

If y is the mean proportional between x and z, prove that

xyz(x + y + z)³ = (xy + yz + zx)³

If $\dfrac{x}{a} =\dfrac{y}{b} = \dfrac{z}{c},$ prove that

$\begin{array}{ll} \text{(i)} & \dfrac{x^3}{a^2} + \dfrac{y^3}{b^2} + \dfrac{z^3}{c^2} = \dfrac{(x + y + z)^3}{(a + b + c)^2} \\ \text{(ii)} & \Big(\dfrac{a^2x^2 + b^2y^2 + c^2z^2}{a^3x + b^3y + c^3z}\Big)^3 = \dfrac{xyz}{abc} \\ \text{(iii)} & \dfrac{ax - by}{(a + b)(x - y)} + \dfrac{by - cz}{(b + c)(y - z)} \ & + \dfrac{cz - ax}{(c + a)(z - x)} = 3. \end{array}$

If $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$, prove that

$\begin{array}{ll} \text{(i)} & (b^2 + d^2 + f^2)(a^2 + c^2 + e^2) \ & = (ab + cd + ef)^2 \\ \text{(ii)} & \dfrac{(a^3 + c^3)^2}{(b^3 + d^3)^2} = \dfrac{e^6}{f^6} \\ \text{(iii)} & \dfrac{a^2}{b^2} + \dfrac{c^2}{d^2} + \dfrac{e^2}{f^2} \ & = \dfrac{ac}{bd} + \dfrac{ce}{df} + \dfrac{ae}{bf} \\ \text{(iv)} & bdf\Big(\dfrac{a + b}{b} + \dfrac{c + d}{d} + \dfrac{e + f}{f}\Big)^3 \ & = 27(a + b)(c + d)(e + f) \end{array}$