If, $\dfrac{x}{b + c - a} = \dfrac{y}{c + a - b} = \dfrac{z}{a + b - c}$, prove that each ratio is equal to $\dfrac{x + y + z}{a + b + c}$.

Also, show that (b − c)x + (c − a)y + (a − b)z = 0.

If b is the mean proportion between a and c, show that:

$\dfrac{a^{4} + a^{2}b^{2} + b^{4}}{b^{4} + b^{2}c^{2} + c^{4}} = \dfrac{a^{2}}{c^{2}}$

If a : b = c : d, prove that (3a + 2b) : (3a − 2b) = (3c + 2d) : (3c − 2d).

If (3a + 5b) : (3a − 5b) = (3c + 5d) : (3c − 5d), prove that a : b = c : d.