Editing Heptagon (section)

===Diagonals and heptagonal triangle===
{{main|Heptagonal triangle}}
[[File:Heptagrams.svg|thumb|100px|''a''=red, ''b''=blue, ''c''=green lines]]

The regular heptagon's side ''a'', shorter [[diagonal#Polygons|diagonal]] ''b'', and longer diagonal ''c'', with ''a''<''b''<''c'', satisfy<ref name=Altintas>Abdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", ''[[Forum Geometricorum]]'' 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf</ref>{{rp|Lemma 1}}

:<math>a^2=c(c-b),</math>
:<math>b^2 =a(c+a),</math>
:<math>c^2 =b(a+b),</math>
:<math>\frac{1}{a}=\frac{1}{b}+\frac{1}{c}</math>  (the [[optic equation]])

and hence

:<math> ab+ac=bc,</math>

and<ref name=Altintas/>{{rp|Coro. 2}}

:<math>b^3+2b^2c-bc^2-c^3=0, </math>
:<math>c^3-2c^2a-ca^2+a^3=0, </math>
:<math>a^3-2a^2b-ab^2+b^3=0,</math>

Thus –''b''/''c'', ''c''/''a'', and ''a''/''b'' all satisfy the [[cubic equation]] <math>t^3-2t^2-t + 1=0.</math> However, no [[algebraic expression]]s with purely real terms exist for the solutions of this equation, because it is an example of [[casus irreducibilis]].

The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by

:<math>b\approx 1.80193\cdot a, \qquad c\approx 2.24698\cdot a.</math>

We also have<ref>Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", ''[[Mathematics Magazine]]'' 46 (1), January 1973, 7–19.</ref>

:<math>b^2-a^2=ac,</math>

:<math>c^2-b^2=ab,</math>

:<math>a^2-c^2=-bc,</math>

and

:<math>\frac{b^2}{a^2}+\frac{c^2}{b^2}+\frac{a^2}{c^2}=5.</math>

A [[heptagonal triangle]] has [[vertex (geometry)|vertices]] coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles <math>\pi/7, 2\pi/7,</math> and <math>4\pi/7.</math> Thus its sides coincide with one side and two particular [[diagonal#Polygons|diagonals]] of the regular heptagon.<ref name=Altintas/>