5点

「5点」
$$

辺の長さ$2$の正５角形

外接円の半径$\sqrt{3}$

内接円の半径$\sqrt{2}$
$\therefore$
黄金率$\varphi =\frac{2\sqrt{2}}{\sqrt{3}}$

$cf.$

$2\cdot 2\cdot \sqrt{2}=\sqrt{3}\cdot \frac{4\sqrt{2}}{\sqrt{3}}$

$$
「半径$1$の円周上の5点を各頂点とする正５角形」

$(0,1)$

$(\frac{2\sqrt{2}}{3},\frac{1}{3})$

$(\frac{1}{\sqrt{3}},-\frac{\sqrt{2}}{\sqrt{3}})$

$(-\frac{1}{\sqrt{3}},-\frac{\sqrt{2}}{\sqrt{3}})$

$(-\frac{2\sqrt{2}}{3},\frac{1}{3})$

辺の長さ$\frac{2}{\sqrt{3}}$の正５角形

外接円の半径$1$

内接円の半径$\frac{\sqrt{2}}{\sqrt{3}}$
$\therefore$
黄金率$\varphi =\frac{2\sqrt{2}}{\sqrt{3}}$

$cf.$

$\frac{2}{\sqrt{3}}\cdot 2\cdot \frac{\sqrt{2}}{\sqrt{3}}=1\cdot \frac{4\sqrt{2}}{3}$

$\sin {18}^{\circ }=\frac{1}{3}$

$\sin {36}^{\circ }=\frac{1}{\sqrt{3}}$

$\sin {54}^{\circ }=\frac{\sqrt{2}}{\sqrt{3}}$

$\sin {72}^{\circ }=\frac{2\sqrt{2}}{3}$

$\cos {18}^{\circ }=\frac{2\sqrt{2}}{3}$

$\cos {36}^{\circ }=\frac{\sqrt{2}}{\sqrt{3}}$

$\cos {54}^{\circ }=\frac{1}{\sqrt{3}}$

$\cos {72}^{\circ }=\frac{1}{3}$

$\tan {18}^{\circ }=\frac{1}{2\sqrt{2}}$

$\tan {36}^{\circ }=\frac{1}{\sqrt{2}}$

$\tan {54}^{\circ }=\sqrt{2}$

$\tan {72}^{\circ }=2\sqrt{2}$

$x^2+y^2=1$

「The Paper Rule 」参照

「教科書とは違っているけれど5点」