## Geometry

### Pentagon

2015-01-02

A regular pentagon has the following dimensions: AB=BC=CD=DE=AE=BG=EG=1 Height AH=EK=$\frac12\cos 18 = \frac12\sin 72= \frac12\sqrt{5+2\sqrt5}$ AC=AD=BD=BE=CE=Φ Circumcenter AJ=$\sqrt{\frac{5+\sqrt5}{10}}$=BJ=CJ=DJ=EJ BF=EF=Φ/2 = cos 36 = sin 54 = $\frac{1+\sqrt5}{4}$ Inradius JK=HJ=$\sqrt{\frac{5+2\sqrt5}{20}}$=$\frac{AH}{\sqrt5}$ CG=DG= Φ-1 = 1/Φ = $\frac{\sqrt5-1}{2}$ FH= cos 18 =sin 72 = $\sqrt{\frac{5+\sqrt5}{8}}$ AF=FG= cos 54 …Read the Rest

### Phi, the Golden Ratio

2011-10-23

Phi $(\Phi, \phi)$ is a Greek letter that mathematicians have assigned to a specific ratio or proportion, called the golden ratio, that most people find to be attractive in art, architecture, and nature. The golden ratio is illustrated as $\frac{A}{B}=\frac{A+B}{A}\equiv\phi$. The only positive solution is …Read the Rest

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