<s>[INST] What are some good questions for this trick, if $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha$ then $\alpha=\frac{a+c+e+...}{b+d+f+...}$? I need some good algebra questions that are applications of this trick, often in a non obvious and elegant way:   $$\text{If } \frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha \text{ then } \alpha=\frac{a+c+e+...}{b+d+f+...}$$
 [/INST] *

*If $\displaystyle\frac a{b+c}=\frac b{c+a}=\frac c{a+b};$  prove that each ratio $\displaystyle=\frac12$ if $\displaystyle a+b+c\ne0$

*If  $\displaystyle\frac{a-b}{x^2}=\frac{b-c}{y^2}=\frac{c-a}{z^2}$ prove that $\displaystyle a=b=c$

*If  $\displaystyle\frac{a-b}{a^2+ab+b^2}=\frac{b-c}{b^2+bc+c^2}=\frac{c-a}{c^2+ca+a^2}$ prove that $\displaystyle a=b=c$ (for a special condition)

*If  $\displaystyle\frac{a+b}{a^2+ab+b^2}=\frac{b+c}{b^2+bc+c^2}$ and $a\ne b\ne c$ prove that each ratio $=\displaystyle\frac{c+a}{c^2+ca+a^2}$
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