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?
1 Answer
$\begingroup$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 $${green+blue\over green} = {green\over blue}$$ $$({green\over blue})^2-{green\over blue}-1=0$$.
$\endgroup$
