Heptagonal triangle explained

In Euclidean geometry, a heptagonal triangle is an obtuse, scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter and longer diagonals of the regular heptagon. All heptagonal triangles are similar (have the same shape), and so they are collectively known as the heptagonal triangle. Its angles have measures

\pi/ 7 , 2 \pi/ 7 , and 4 \pi/ 7 , and it is the only triangle with angles in the ratios 1:2:4. The heptagonal triangle has various remarkable properties.

Key points

The heptagonal triangle's nine-point center is also its first Brocard point.^[1]

The second Brocard point lies on the nine-point circle.^[2]

The circumcenter and the Fermat points of a heptagonal triangle form an equilateral triangle.^[1]

The distance between the circumcenter O and the orthocenter H is given by^[2]

OH = R\sqrt{ 2 },

where R is the circumradius. The squared distance from the incenter I to the orthocenter is^[2]

IH ^{2= (R ^{2+4r} ^{2})/(2)} ,

where r is the inradius.

The two tangents from the orthocenter to the circumcircle are mutually perpendicular.^[2]

Relations of distances

Sides

The heptagonal triangle's sides a < b < c coincide respectively with the regular heptagon's side, shorter diagonal, and longer diagonal. They satisfy^[3]

\begin{align} a ^{2} & = c(c - b), \\[ 5 pt] b ^{2} & = a(c + a), \\[ 5 pt] c ^{2} & = b(a + b), \\[ 5 pt] (1)/(a) & = (1)/(b) + (1)/(c) \end{align}

(the latter^[2] being the optic equation) and hence

ab + ac = bc,

and^[3]

b ^{3+2b} ^{2c-bc} ^{2-c} ^{3=0,} c ^{3-2c} ^{2a-ca} ^{2+a} ^{3=0,} a ^{3-2a} ^{2b-ab} ^{2+b} ^{3=0.}

Thus –b/c, c/a, and a/b all satisfy the cubic equation

t ^{3-2t} ^{2-t} + 1=0 .

However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis.

The approximate relation of the sides is

b ≈ 1 . 80193 ⋅ a, c ≈ 2 . 24698 ⋅ a.

We also have^{[4] [5]}

(a ^{2})/(bc) , - (b ^{2})/(ca) , - (c ^{2})/(ab) satisfy the cubic equationt ^{3+4t} ^{2+3t-1=0.}

We also have^[4]

(a ^{3})/(bc ^{2}) , - (b ^{3})/(ca ^{2}) , (c ^{3})/(ab ^{2}) satisfy the cubic equationt ^{3-t} ^{2-9t+1=0.}

We also have^[4]

(a ^{3})/(b ^{2c}) , (b ^{3})/(c ^{2a}) , - (c ^{3})/(a ^{2b}) satisfy the cubic equationt ^{3+5t} ^{2-8t+1=0.}

We also have^[2]

b ^{2-a} ^{2=ac,} c ^{2-b} ^{2=ab,} a ^{2-c} ^{2=-bc,}

and^[2]

(b ^{2})/(+ a ^{2}) (c ^{2})/(+ b ^{2}) (a ^{2})/(c ^{2}) =5 .

We also have^[4]

ab - bc + ca =0 , a ^{3} b - b ^{3} c + c ^{3} a =0 , a ^{4} b + b ^{4} c - c ^{4} a =0 , a ^{11} b ^{3} - b ^{11} c ^{3} + c ^{11} a ^{3} =0 .

Altitudes

The altitudes h_a, h_b, and h_c satisfy

h _{a=h} _{b+h} _{c} ^[2]

and

h ^{2= (a ^{2+b} ^{2+c} ^{2})/(2)} _{c} . ^[2]

The altitude from side b (opposite angle B) is half the internal angle bisector

w _{A} of A:^[2] 2 h _{b=w} _{A.}

Here angle A is the smallest angle, and B is the second smallest.

Internal angle bisectors

w _{A,} w _{B,} and w _{C} of angles A, B, and C respectively:^[2] w _{A=b+c,} w _{B=c-a,} w _{C=b-a.}

Circumradius, inradius, and exradius

The triangle's area is

A = (\sqrt{ 7)/() }R^2,

where R is the triangle's circumradius.

We have^[2]

a ^{2+b} ^{2+c} ^{2=7R} ^{2.}

We also have^[6]

a ^{4+b} ^{4+c} ^{4=21R} ^{4.} a ^{6+b} ^{6+c} ^{6=70R} ^{6.}

The ratio r /R of the inradius to the circumradius is the positive solution of the cubic equation

8 x ^{3+28x} ^{2+14x-7=0.}

In addition,^[2]

(1)/(+ a ^{2}) (1)/(+ b ^{2}) (1)/(= c ^{2}) (2)/(R ^{2}) .

We also have^[6]

(1)/(+ a ^{4}) (1)/(+ b ^{4}) (1)/(= c ^{4}) (2)/(R ^{4}) . (1)/(+ a ^{6}) (1)/(+ b ^{6}) (1)/(= c ^{6}) (17)/(7 R ^{6}) .

In general for all integer n,

a ^{2n} + b ^{2n} + c ^{2n} = g(n)( 2 R) ^{2n} where g( -1 ) = 8 , g( 0 ) =3 , g( 1 ) =7 and g(n) =7 g(n -1 ) -14 g(n -2 ) +7 g(n -3 ).

We also have^[6]

2 b ^{2-a} ^{2=\sqrt{7}bR,} 2 c ^{2-b} ^{2=\sqrt{7}cR,} 2 a ^{2-c} ^{2=-\sqrt{7}aR.}

We also have^[4]

a ^{3} c + b ^{3} a - c ^{3} b = -7 R ^{4} , a ^{4} c - b ^{4} a + c ^{4} b = 7 \sqrt{ 7 }R ^{5} , a ^{11} c ^{3} + b ^{11} a ^{3} - c ^{11} b ^{3} = -7 ^{3} 17 R ^{14} .

The exradius r_a corresponding to side a equals the radius of the nine-point circle of the heptagonal triangle.^[2]

Orthic triangle

The heptagonal triangle's orthic triangle, with vertices at the feet of the altitudes, is similar to the heptagonal triangle, with similarity ratio 1:2. The heptagonal triangle is the only obtuse triangle that is similar to its orthic triangle (the equilateral triangle being the only acute one).^[2]

Trigonometric properties

Trigonometric identities

The various trigonometric identities associated with the heptagonal triangle include these:^{[2] [7] [6] [4]}

\sin^2\!A \!&\! \times \!&\! \sin^2\!B \!&\! \times \!&\! \sin^2\!C \!&\! = \!&\! \frac \\[2pt] \sin^2\!A \!&\! + \!&\! \sin^2\!B \!&\! + \!&\! \sin^2\!C \!&\! = \!&\! \frac \\[2pt] \cos^2\!A \!&\! + \!&\! \cos^2\!B \!&\! + \!&\! \cos^2\!C \!&\! = \!&\! \frac \\[2pt] \tan^2\!A \!&\! + \!&\! \tan^2\!B \!&\! + \!&\! \tan^2\!C \!&\! = \!&\! 21 \\[2pt] \sec^2\!A \!&\! + \!&\! \sec^2\!B \!&\! + \!&\! \sec^2\!C \!&\! = \!&\! 24 \\[2pt] \csc^2\!A \!&\! + \!&\! \csc^2\!B \!&\! + \!&\! \csc^2\!C \!&\! = \!&\! 8 \\[2pt] \cot^2\!A \!&\! + \!&\! \cot^2\!B \!&\! + \!&\! \cot^2\!C \!&\! = \!&\! 5 \\[8pt]

\sin^4\!A \!&\! + \!&\! \sin^4\!B \!&\! + \!&\! \sin^4\!C \!&\! = \!&\! \frac \\[2pt] \cos^4\!A \!&\! + \!&\! \cos^4\!B \!&\! + \!&\! \cos^4\!C \!&\! = \!&\! \frac \\[2pt] \sec^4\!A \!&\! + \!&\! \sec^4\!B \!&\! + \!&\! \sec^4\!C \!&\! = \!&\! 416 \\[2pt] \csc^4\!A \!&\! + \!&\! \csc^4\!B \!&\! + \!&\! \csc^4\!C \!&\! = \!&\! 32 \\[8pt]\end

^[6]

^[4]

^[4]

$$\backslash sin\; A\backslash sin\; B\; -\; \backslash sin\; B\backslash sin\; C\; +\; \backslash sin\; C\backslash sin\; A\; =\; 0$$ ^[8]

Cubic polynomials

64 y ^{3-112y} ^{2+56y-7=0} has solutions^[2] \sin ^{2} A, \sin ^{2} B, \sin ^{2} C.

The positive solution of the cubic equation

x ^{3+x} ^{2-2x-1=0} equals 2 \cos B. ^[9]

The roots of the cubic equation

x ^{3} - \tfrac{\sqrt 7 }{ 2 }x ^{2} + \tfrac{\sqrt 7 }{ 8 } = 0 are^[4] \sin 2 A, \sin 2 B, \sin 2 C.

The roots of the cubic equation

x ^{3} - \tfrac{\sqrt 7 }{ 2 } x ^{2} + \tfrac{\sqrt 7 }{ 8 } = 0 are - \sin A, \sin B, \sin C.

The roots of the cubic equation

x ^{3} + \tfrac{ 1 }{ 2 }x ^{2} - \tfrac{ 1 }{ 2 }x - \tfrac{ 1 }{ 8 } = 0 are - \cos A, \cos B, \cos C.

The roots of the cubic equation

x ^{3} + \sqrt{ 7 }x ^{2} - 7 x + \sqrt{ 7 } = 0 are \tan A, \tan B, \tan C. The roots of the cubic equation x ^{3} - 21 x ^{2} + 35 x - 7 = 0 are \tan ^{2} A, \tan ^{2} B, \tan ^{2} C.

Sequences

For an integer, let

Value of :	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
S(n)	3	\tfrac{\sqrt 7 }{ 2 }	\tfrac{ 7 }{ 2 ^{2}}	\tfrac{\sqrt 7 }{ 2 }	\tfrac{ 7 ⋅ 3 }{ 2 ^{4}}	\tfrac{ 7 \sqrt 7 }{ 2 ^{4}}	\tfrac{ 7 ⋅ 5 }{ 2 ^{5}}	\tfrac{ 7 ^{2\sqrt} 7 }{ 2 ^{7}}	\tfrac{ 7 ^{2 ⋅ 5}{2} ^{8}}	\tfrac{ 7 ⋅ 25 \sqrt 7 }{ 2 ^{9}}	\tfrac{ 7 ^{2 ⋅ 9}{2} ^{9}}	\tfrac{ 7 ^{2 ⋅ 13\sqrt} 7 }{ 2 ^{11} }	\tfrac{ 7 ^{2 ⋅ 33}{2} ^{11} }	\tfrac{ 7 ^{2 ⋅ 3\sqrt} 7 }{ 2 ^{9}}	\tfrac{ 7 ^{4 ⋅ 5}{2} ^{14} }	\tfrac{ 7 ^{2 ⋅ 179\sqrt} 7 }{ 2 ^{15} }	\tfrac{ 7 ^{3 ⋅ 131}{2} ^{16} }	\tfrac{ 7 ^{3 ⋅ 3\sqrt} 7 }{ 2 ^{12} }	\tfrac{ 7 ^{3 ⋅ 493}{2} ^{18} }	\tfrac{ 7 ^{3 ⋅ 181\sqrt} 7 }{ 2 ^{18} }	\tfrac{ 7 ^{5 ⋅ 19}{2} ^{19} }
S( - n)	3	0	2 ^{3}	- \tfrac{ 2 ^{3 ⋅ 3\sqrt} 7 }{ 7 }	2 ^{5}	- \tfrac{ 2 ^{5 ⋅ 5\sqrt} 7 }{ 7 }	\tfrac{ 2 ^{6 ⋅ 17}{7}}	-2 ^{7\sqrt{7}}	\tfrac{ 2 ^{9 ⋅ 11}{7}}	- \tfrac{ 2 ^{10} ⋅ 33 \sqrt 7 }{ 7 ^{2}}	\tfrac{ 2 ^{10} ⋅ 29 }{ 7 }	- \tfrac{ 2 ^{14} ⋅ 11 \sqrt 7 }{ 7 ^{2}}	\tfrac{ 2 ^{12} ⋅ 269 }{ 7 ^{2}}	- \tfrac{ 2 ^{13} ⋅ 117 \sqrt 7 }{ 7 ^{2}}	\tfrac{ 2 ^{14} ⋅ 51 }{ 7 }	- \tfrac{ 2 ^{21} ⋅ 17 \sqrt 7 }{ 7 ^{3}}	\tfrac{ 2 ^{17} ⋅ 237 }{ 7 ^{2}}	- \tfrac{ 2 ^{17} ⋅ 1445 \sqrt 7 }{ 7 ^{3}}	\tfrac{ 2 ^{19} ⋅ 2203 }{ 7 ^{3}}	- \tfrac{ 2 ^{19} ⋅ 1919 \sqrt 7 }{ 7 ^{3}}	\tfrac{ 2 ^{20} ⋅ 5851 }{ 7 ^{3}}
C(n)	3	- \tfrac{ 1 }{ 2 }	\tfrac{ 5 }{ 4 }	- \tfrac{ 1 }{ 2 }	\tfrac{ 13 }{ 16 }	- \tfrac{ 1 }{ 2 }	\tfrac{ 19 }{ 32 }	- \tfrac{ 57 }{ 128 }	\tfrac{ 117 }{ 256 }	- \tfrac{ 193 }{ 512 }	\tfrac{ 185 }{ 512 }
C( - n)	3	-4	24	-88	416	-1824	8256	-36992	166400	-747520	3359744
T(n)	3	- \sqrt{ 7 }	7 ⋅ 3	-31 \sqrt{ 7 }	7 ⋅ 53	-7 ⋅ 87 \sqrt{ 7 }	7 ⋅ 1011	-7 ^{2 ⋅ 239\sqrt{7}}	7 ^{2 ⋅ 2771}	-7 ⋅ 32119 \sqrt{ 7 }	7 ^{2 ⋅ 53189}
T( - n)	3	\sqrt{ 7 }	5	\tfrac{ 25 \sqrt 7 }{ 7 }	19	\tfrac{ 103 \sqrt 7 }{ 7 }	\tfrac{ 563 }{ 7 }	7 ⋅ 9 \sqrt{ 7 }	\tfrac{ 2421 }{ 7 }	\tfrac{ 13297 \sqrt 7 }{ 7 ^{2}}	\tfrac{ 10435 }{ 7 }

Ramanujan identities

We also have Ramanujan type identities,^{[6] [10]}

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! -\sqrt[18] \times \sqrt[3] \\[2pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! \sqrt[18] \times \sqrt[3] \\[6pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! \sqrt[3] \\[8pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! \sqrt[3] \\[6pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! -\sqrt[18] \times \sqrt[3] \\[2pt] \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! \sqrt[18] \times \sqrt[3]\end

\frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \frac \times \sqrt[3] \\[2pt]

\frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \sqrt[3] \\[6pt]

\frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \sqrt[3] \\[2pt]

\frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! -\frac \times \sqrt[3] \\[2pt]

\frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \frac \times \sqrt[3] \end

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! 0 \\[2pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! -\frac \\[2pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! 0 \\[2pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! -3\times \frac \\[2pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! 0 \\[2pt]

\sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! + \!&\! \sqrt[3] \!&\! = \!&\! -61\times \frac. \end^[8]

$$$$

Notes and References

1. Paul Yiu, "Heptagonal Triangles and Their Companions", Forum Geometricorum 9, 2009, 125–148. http://forumgeom.fau.edu/FG2009volume9/FG200912.pdf
2. Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", Mathematics Magazine 46 (1), January 1973, 7–19.
3. Abdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", Forum Geometricorum 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf
4. Wang, Kai. “Heptagonal Triangle and Trigonometric Identities”, Forum Geometricorum 19, 2019, 29–38.
5. Wang, Kai.https://www.researchgate.net/publication/335392159_On_cubic_equations_with_zero_sums_of_cubic_roots_of_roots
6. Wang, Kai. https://www.researchgate.net/publication/327825153_Trigonometric_Properties_For_Heptagonal_Triangle
7. Web site: Weisstein . Eric W. . Heptagonal Triangle . 2024-08-02 . mathworld.wolfram.com . en.
8. Wang, Kai.https://www.researchgate.net/publication/336813631_Topics_of_Ramanujan_type_identities_for_PI7
9. Gleason. Andrew Mattei. Angle trisection, the heptagon, and the triskaidecagon . The American Mathematical Monthly. March 1988. 95. 3 . 185–194. https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf#3 . 10.2307/2323624. 2015-12-19 . dead.
10. Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007).