Heptagonal Triangle

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.Paul Yiu, "Heptagonal Triangles and Their Companions", ''Forum Geometricorum'' 9, 2009, 125–148. http://forumgeom.fau.edu/FG2009volume9/FG200912.pdf

The second Brocard point lies on the nine-point circle.Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", ''Mathematics Magazine'' 46 (1), January 1973, 7–19.

The circumcenter and the Fermat points of a heptagonal triangle form an equilateral triangle.

The distance between the circumcenter ''O'' and the orthocenter ''H'' is given by

:$OH=R\sqrt,$

where ''R'' is the circumradius. The squared distance from the incenter ''I'' to the orthocenter is

:$IH^2=\frac,$

where ''r'' is the inradius.

The two tangents from the orthocenter to the circumcircle are mutually perpendicular.

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 satisfyAbdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", ''Forum Geometricorum'' 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf

:
\begin
a^2 & =c(c-b), \\ ptb^2 & =a(c+a), \\ ptc^2 & =b(a+b), \\ pt\frac 1 a & =\frac 1 b + \frac 1 c
\end

(the latter being the optic equation) and hence

:$ab+ac=bc,$

and

:$b^3+2b^2c-bc^2-c^3=0,$
:$c^3-2c^2a-ca^2+a^3=0,$
:$a^3-2a^2b-ab^2+b^3=0.$

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

:$b\approx 1.80193\cdot a, \qquad c\approx 2.24698\cdot a.$

We also haveWang, Kai. “Heptagonal Triangle and Trigonometric Identities”, ''Forum Geometricorum'' 19, 2019, 29–38.Wang, Kai.
https://www.researchgate.net/publication/335392159_On_cubic_equations_with_zero_sums_of_cubic_roots_of_roots

:$\frac, \quad -\frac, \quad -\frac$
satisfy the cubic equation
:$t^3+4t^2+3t-1=0.$

We also have
:$\frac, \quad -\frac, \quad \frac$
satisfy the cubic equation
:$t^3-t^2-9t+1=0.$

We also have
:$\frac, \quad \frac, \quad -\frac$
satisfy the cubic equation
:$t^3+5t^2-8t+1=0.$

We also have

:$b^2-a^2=ac,$

:$c^2-b^2=ab,$

:$a^2-c^2=-bc,$

and

:$\frac+\frac+\frac=5.$

We also have
:$ab-bc+ca=0,$
:$a^b-b^c+c^a=0,$
:$a^b+b^c-c^a=0,$
:$a^b^-b^c^+c^a^=0.$

There are no other (''m, n''), ''m, n'' > 0, ''m, n'' < 2000 such that
:$a^b^ \pm b^c^ \pm c^a^=0.$

Altitudes

The altitudes ''h''_''a'', ''h''_''b'', and ''h''_''c'' satisfy

:$h_a=h_b+h_c$

and

:$h_a^2+h_b^2+h_c^2=\frac.$

The altitude from side ''b'' (opposite angle ''B'') is half the internal angle bisector $w_A$ of ''A'':

:$2h_b=w_A.$

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

Internal angle bisectors

We have these properties of the internal angle bisectors $w_A, w_B,$ and $w_C$ of angles ''A, B'', and ''C'' respectively:

:$w_A=b+c,$

:$w_B=c-a,$

:$w_C=b-a.$

Circumradius, inradius, and exradius

The triangle's area is

:$A=\fracR^2,$

where ''R'' is the triangle's circumradius.

We have

:$a^2+b^2+c^2=7R^2.$

We also haveWang, Kai. https://www.researchgate.net/publication/327825153_Trigonometric_Properties_For_Heptagonal_Triangle
:$a^4+b^4+c^4=21R^4.$
:$a^6+b^6+c^6=70R^6.$

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

:$8x^3+28x^2+14x-7=0.$

In addition,

:$\frac+\frac+\frac=\frac.$

We also have
:$\frac+\frac+\frac=\frac.$
:$\frac+\frac+\frac=\frac.$

In general for all integer ''n'',
:$a^+b^+c^=g(n)(2R)^$
where
:$g(-1) = 8, \quad g(0)=3, \quad g(1)=7$
and
:$g(n)=7g(n-1)-14g(n-2)+7g(n-3).$

We also have
:$2b^2-a^2=\sqrtbR, \quad 2c^2-b^2=\sqrtcR, \quad 2a^2-c^2=-\sqrtaR.$

We also have
:$a^c + b^a - c^b = -7R^,$
:$a^c - b^a + c^b = 7\sqrtR^,$
:$a^c^+b^a - c^b^ = -7^17R^.$

The exradius ''r''_''a'' corresponding to side ''a'' equals the radius of the nine-point circle of the heptagonal triangle.

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).

Trigonometric properties

The various trigonometric identities associated with the heptagonal triangle include these:Weisstein, Eric W. "Heptagonal Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HeptagonalTriangle.html

:$A=\frac, \quad B=\frac, \quad C=\frac.$

:$\cos A=b/2a, \quad \cos B=c/2b, \quad \cos C =-a/2c,$

:$\cos A \cos B \cos C = -\frac,$
:$\cos^2A+\cos^2B+\cos^2C = \frac,$
:$\cos^4A+\cos^4B+\cos^4C=\frac,$
:$\cot A+\cot B+\cot C=\sqrt,$
:$\cot^2A+\cot^2B+\cot^2C=5,$
:$\csc^2A+\csc^2B+\csc^2C=8,$
:$\csc^4A+\csc^4B+\csc^4C=32,$
:$\sec^2A+\sec^2B+\sec^2C=24,$
:$\sec^4A+\sec^4B+\sec^4C=416,$
:$\sin A\sin B\sin C =\frac,$
:$\sin^2A\sin^2B\sin^2C=\frac,$
:$\sin^2A+\sin^2B+\sin^2C=\frac,$
:$\sin^4A+\sin^4B+\sin^4C=\frac,$
:$\tan A\tan B\tan C=\tan A+\tan B+\tan C=-\sqrt,$
:$\tan^2A+\tan^2B+\tan^2C=21.$

The cubic equation

:$64y^3-112y^2+56y-7=0$

has solutions $\sin^2\frac, \sin^2\frac,$ and $\sin^2\frac,$ which are the squared sines of the angles of the triangle.

The positive solution of the cubic equation

:$x^3+x^2-2x-1=0$

equals $2\cos\frac,$ which is twice the cosine of one of the triangle’s angles.

Sin (2π / 7), sin (4π / 7), and sin (8π / 7) are the roots of

:$x^3-\fracx^2+ \frac=0.$

We also have:
:$\sin A - \sin B - \sin C = - \frac,$
:$\sin A\sin B - \sin B\sin C + \sin C\sin A = 0,$
:$\sin A\sin B\sin C = \frac.$

:$-\sin A, \sin B, \sin C \text x^3 - \frac x^2 + \frac = 0.$

For an integer ''n'' , let
:$S(n) = (-\sin)^ + \sin^ + \sin^ .$

For ''n'' = 0,...,20,
:$S(n) = 3, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac,$
:$\frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac, \frac.$

For ''n''= 0, -1, ,..-20,
:$S(n) = 3, 0, 2^, -\frac, 2^, -\frac, \frac, -2^\sqrt, \frac, -\frac, \frac, -\frac, \frac,$
:$-\frac, \frac, -\frac, \frac, -\frac, \frac, -\frac, \frac.$

:$-\cos A, \cos B, \cos C \text x^3 + \frac x^2 - \frac x- \frac= 0.$

For an integer ''n'' , let
:$C(n) = (-\cos)^ + \cos^ + \cos^ .$

For ''n''= 0, 1, ,..10,
:$C(n) = 3, -\frac, \frac, -\frac, \frac, -\frac, \frac, -\frac, \frac, -\frac, \frac,...$
:$C(-n) = 3, -4, 24, -88, 416, -1824, 8256, -36992, 166400, -747520, 3359744,...$

:$\tan A, \tan B, \tan C \text x^3 + \sqrt x^2 - 7x + \sqrt = 0.$

:$\tan ^2 A, \tan^2 B, \tan^2 C \text x^3 - 21 x^2 + 35x - 7 = 0.$

For an integer ''n'' , let
:$T(n) = \tan^ + \tan^ + \tan^ .$

For ''n''= 0, 1, ,..10,
:$T(n) = 3, -\sqrt, 7\cdot3, -31\sqrt, 7\cdot53, -7\cdot87\sqrt, 7\cdot1011, -7^\cdot239\sqrt, 7^\cdot2771, -7\cdot32119\sqrt, 7^\cdot53189,$
:$T(-n) = 3, \sqrt, 5, \frac, 19, \frac, \frac, 7\cdot9\sqrt, \frac, \frac, \frac,...$

We also haveVictor Hugo Moll, An elementary trigonometric equation, https://arxiv.org/abs/0709.3755, 2007
:$\tan A - 4\sin B = - \sqrt,$
:$\tan B - 4\sin C = - \sqrt,$
:$\tan C + 4\sin A = - \sqrt.$

We also have
:$\cot^ A = 1 - \frac,$
:$\cot^ B = 1 - \frac,$
:$\cot^ C = 1 - \frac.$

We also have
:$\cos A = -\frac + \frac \sin^ C,$
:$\cos^ A = \frac + \frac \sin^ A,$
:$\cot A = \frac + \frac \cos B,$
:$\cot^ A = 3 + \frac \sin A,$
:$\cot A = \sqrt + \frac \sin^ B,$
:$\csc^ A = -\frac + \frac \tan^2 C,$
:$\sec A = 2 + 4 \cos C,$
:$\sec A = 6 - 8 \sin^ B,$
:$\sec A = 4 - \frac \sin^ B,$
:$\sin^ A = \frac + \frac \cos B,$
:$\sin^ A = -\frac + \frac \cos B,$

We also haveWang, Kai.
https://www.researchgate.net/publication/336813631_Topics_of_Ramanujan_type_identities_for_PI7
:$\sin^B\sin C-\sin^C\sin A-\sin^A\sin B = 0,$
:$\sin B\sin^C-\sin C\sin^A-\sin A\sin^B = \frac,$
:$\sin^B\sin C-\sin^C\sin A+\sin^A\sin B = 0,$
:$\sin B\sin^C+\sin C\sin^A-\sin A\sin^B = \frac,$
:$\sin^B\sin^C-\sin^C\sin^A-\sin^A\sin^B = 0,$
:$\sin^B\sin^C-\sin^C\sin^A-\sin^A\sin^B = \frac.$

We also have Ramanujan type identities,Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007).

:
\sqrt + \sqrt +\sqrt =

:
\text \left(-\sqrt 8right)
\sqrt

:$\frac + \frac + \frac =$

:
\text \left(-\frac\right)
\sqrt

:
\sqrt
+ \sqrt
+ \sqrt
=

:
\text \left(\sqrt 8right)
\sqrt

:$\frac +\frac +\frac =$

:
\text \left(\frac\right)
\sqrt

:
\sqrt
+\sqrt
+\sqrt
=
\sqrt
:
\frac
+\frac
+\frac
=
\sqrt

:
\sqrt
+\sqrt
+\sqrt =
\sqrt
:
\frac
+\frac
+\frac =
\sqrt

:
\sqrt
+\sqrt
+\sqrt
=

:
\text \left(-\sqrt 8right)
\sqrt
:$\frac +\frac +\frac =$
:
\text \left(-\frac\right)
\sqrt

:
\sqrt
+\sqrt +\sqrt =

:
\text \left(\sqrt 8right)
\sqrt

:$\frac +\frac +\frac =$
:
\text \left(\frac\right)
\sqrt

We also have
:
\sqrt + \sqrt
+ \sqrt
= -\sqrt

:
\sqrt
+ \sqrt
+ \sqrt = 0.

:
\sqrt + \sqrt +\sqrt
\left(-\sqrt 8right)
\sqrt

:
\sqrt
+ \sqrt
+ \sqrt = -\sqrt 2.

:
\sqrt
+ \sqrt
+ \sqrt = 0.

:
\sqrt
+ \sqrt
+ \sqrt = -3*\sqrt 2.

:
\sqrt
+ \sqrt
+ \sqrt = 0.

:
\sqrt
+ \sqrt
+ \sqrt = -61*\sqrt 8.

References

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Types of triangles