If b is the mean proportional between a and c , prove that a, c, a² + b² and b² + c² are proportional.

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 a + c = mb and $\dfrac{1}{b} + \dfrac{1}{d} = \dfrac{m}{c}$, prove that a, b, c and d are in proportion.

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

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