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

  Regular heptagon
   Longer diagonals
  Shorter diagonals
Each of the fourteen congruent heptagonal triangles has one green side, one blue side, and one red side.

Key points

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The heptagonal triangle's nine-point center is also its first Brocard point.[1]: Propos. 12 

The second Brocard point lies on the nine-point circle.[2]: p. 19 

The circumcenter and the Fermat points of a heptagonal triangle form an equilateral triangle.[1]: Thm. 22 

The distance between the circumcenter O and the orthocenter H is given by[2]: p. 19 

 

where R is the circumradius. The squared distance from the incenter I to the orthocenter is[2]: p. 19 

 

where r is the inradius.

The two tangents from the orthocenter to the circumcircle are mutually perpendicular.[2]: p. 19 

Relations of distances

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Sides

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The heptagonal triangle's sides a < b < c coincide respectively with the regular heptagon's side, shorter diagonal, and longer diagonal. They satisfy[3]: Lemma 1 

 

(the latter[2]: p. 13  being the optic equation) and hence

 

and[3]: Coro. 2 

 
 
 

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

 

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

 

We also have[4][5]

 

satisfy the cubic equation

 

We also have[4]

 

satisfy the cubic equation

 

We also have[4]

 

satisfy the cubic equation

 

We also have[2]: p. 14 

 
 
 

and[2]: p. 15 

 

We also have[4]

 
 
 
 

Altitudes

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The altitudes ha, hb, and hc satisfy

 [2]: p. 13 

and

 [2]: p. 14 

The altitude from side b (opposite angle B) is half the internal angle bisector   of A:[2]: p. 19 

 

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

Internal angle bisectors

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We have these properties of the internal angle bisectors   and   of angles A, B, and C respectively:[2]: p. 16 

 
 
 

Circumradius, inradius, and exradius

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The triangle's area is[6]

 

where R is the triangle's circumradius.

We have[2]: p. 12 

 

We also have[7]

 
 

The ratio r /R of the inradius to the circumradius is the positive solution of the cubic equation[6]

 

In addition,[2]: p. 15 

 

We also have[7]

 
 

In general for all integer n,

 

where

 

and

 

We also have[7]

 

We also have[4]

 
 
 

The exradius ra corresponding to side a equals the radius of the nine-point circle of the heptagonal triangle.[2]: p. 15 

Orthic triangle

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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]: pp. 12–13 

Hyperbola

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The rectangular hyperbola through   has the following properties:

  • first focus  
  • center   is on Euler circle (general property) and on circle  
  • second focus   is on the circumcircle

Trigonometric properties

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Trigonometric identities

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The various trigonometric identities associated with the heptagonal triangle include these:[2]: pp. 13–14 [6][7]

 [4]: Proposition 10 

 

 [7][8]

 [4]

 [4]

     [9]

Cubic polynomials

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The cubic equation   has solutions[2]: p. 14   

The positive solution of the cubic equation   equals  [10]: p. 186–187 

The roots of the cubic equation   are[4]  

The roots of the cubic equation   are  

The roots of the cubic equation   are  

The roots of the cubic equation   are  

The roots of the cubic equation   are  

Sequences

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For an integer n, let  

Value of n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
                                           
                                           
                       
                       
                       
                       

Ramanujan identities

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We also have Ramanujan type identities,[7][11]

 

 

 [9]

References

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  1. ^ a b Yiu, Paul (2009). "Heptagonal Triangles and Their Companions" (PDF). Forum Geometricorum. 9: 125–148.
  2. ^ a b c d e f g h i j k l m n o p q Bankoff, Leon; Garfunkel, Jack (1973). "The Heptagonal Triangle". Mathematics Magazine. 46 (1): 7–19. doi:10.2307/2688574. JSTOR 2688574.
  3. ^ a b Altintas, Abdilkadir (2016). "Some Collinearities in the Heptagonal Triangle" (PDF). Forum Geometricorum. 16: 249–256.
  4. ^ a b c d e f g h i Wang, Kai (2019). "Heptagonal Triangle and Trigonometric Identities". Forum Geometricorum. 19: 29–38.
  5. ^ Wang, Kai (August 2019). "On cubic equations with zero sums of cubic roots of roots" – via ResearchGate.
  6. ^ a b c Weisstein, Eric W. "Heptagonal Triangle". mathworld.wolfram.com. Retrieved 2024-08-02.
  7. ^ a b c d e f Wang, Kai (September 2018). "Trigonometric Properties For Heptagonal Triangle" – via ResearchGate.
  8. ^ Moll, Victor H. (2007-09-24). "An elementary trigonometric equation". arXiv:0709.3755 [math.NT].
  9. ^ a b Wang, Kai (October 2019). "On Ramanujan Type Identities For PI/7" – via ResearchGate.
  10. ^ Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. JSTOR 2323624. Archived from the original (PDF) on 2015-12-19.
  11. ^ Witula, Roman; Slota, Damian (2007). "New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7" (PDF). Journal of Integer Sequences. 10 (5) 07.5.6. Bibcode:2007JIntS..10...56W.