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Decagon
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== The golden ratio in decagon == Both in the construction with given circumcircle<ref name="Henry Green">{{citation|title=Euclid's Plane Geometry, Books III–VI, Practically Applied, or Gradations in Euclid, Part II|first=Henry|last=Green|publisher=London: Simpkin, Marshall,& CO.| year=1861|page=116|url=https://books.google.com/books?id=DjQDAAAAQAAJ&q=II.+Practically+decagon+Euklid+p.+116&pg=PA116}}. Retrieved 10 February 2016.</ref> as well as with given side length is the [[Golden ratio#Geometry|golden ratio dividing a line segment by exterior division]] the determining construction element. * In the construction with given circumcircle the circular arc around G with radius {{Overline|GE<sub>3</sub>}} produces the segment {{Overline|AH}}, whose division corresponds to the golden ratio. :<math>\frac{\overline{AM}}{\overline{MH}} = \frac{\overline{AH}}{\overline{AM}} = \frac{1+ \sqrt{5}}{2} = \Phi \approx 1.618 \text{.}</math> * In the construction with given side length<ref name="Jürgen Köller">{{citation|title=Regelmäßiges Zehneck, → 3. Section "Formeln, Ist die Seite a gegeben ..." |first=Jürgen|last=Köller| year=2005|url=http://www.mathematische-basteleien.de/zehneck.htm|language=de}}. Retrieved 10 February 2016.</ref> the circular arc around D with radius {{Overline|DA}} produces the segment {{Overline|E<sub>10</sub>F}}, whose division corresponds to the [[Golden ratio#Calculation|golden ratio]]. :<math>\frac{\overline{E_1 E_{10}}}{\overline{E_1 F}} = \frac{\overline{E_{10} F}}{\overline{E_1 E_{10}}} = \frac{R}{a} = \frac{1+ \sqrt{5}}{2} =\Phi \approx 1.618 \text{.}</math> <div class="skin-invert-image"> {{multiple image | align = left | image1 = 01-Zehneck-Animation.gif | width1 = 287 | alt1 = | caption1 = Decagon with given circumcircle,<ref name="Henry Green" /> animation | image2 = 01-Zehneck-Seitenlänge-Animation.gif | width2 = 300 | alt2 = | caption2 = Decagon with a given side length,<ref name="Jürgen Köller" /> animation | footer = }}</div> {{clear}}
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