Question

If $$a : b :: c : d$$, show that $$\frac{a^2 + ab + b^2}{a^2 - ab + b^2} = \frac{c^2 + cd + d^2}{c^2 - cd + d^2}$$.

Answer 1

Let $$\frac{a}{b} = \frac{c}{d} = k$$, so $$a = bk$$ and $$c = dk$$.

$$\text{LHS} = \frac{b^2k^2 + b^2k + b^2}{b^2k^2 - b^2k + b^2}$$

$$= \frac{k^2 + k + 1}{k^2 - k + 1}$$

$$\text{RHS} = \frac{d^2k^2 + d^2k + d^2}{d^2k^2 - d^2k + d^2}$$

$$= \frac{k^2 + k + 1}{k^2 - k + 1}$$

$$\text{LHS} = \text{RHS}$$