Pythagoreans used the pentagram as their mystical symbol. They believed that every number-shape had a hidden meaning and the pentagram is related with the golden ratio. My questions are:
- The only reason why the golden ration is so famous is because of this relationships:
$\frac{\text{red}}{\text{green}}=\frac{\text{green}}{\text{blue}}=\frac{\text{blue}}{\text{purple}}=\varphi ??$
- How can it be proved without using angles?
It can be proved using similar triangles. The purple triangle and green triangle are similar because all sides are either on the same line or parallel. The purple triangle and triangle $ABC$ are similar for obvious reason, giving ${red\over green}={green\over blue}={blue \over purple}$.
For the ratio, notice that $green+blue=red$ and ${red\over green}={green\over blue}$ so $${green+blue\over green} = {green\over blue}$$ $$({green\over blue})^2-{green\over blue}-1=0$$.