Heptagonal triangle


A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon. Thus its sides coincide with one side and the adjacent shorter and longer diagonals of the regular heptagon. All heptagonal triangles are similar, 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

Key points

The heptagonal triangle's nine-point center is also its first Brocard point.
The second Brocard point lies on the nine-point circle.
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
where R is the circumradius. The squared distance from the incenter I to the orthocenter is
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 satisfy
and hence
and
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
satisfy the cubic equation
We also have
satisfy the cubic equation
We also have
satisfy the cubic equation
We also have
and
We also have
There are no other, m, n > 0, m, n < 2000 such that

Altitudes

The altitudes ha, hb, and hc satisfy
and
The altitude from side b is half the internal angle bisector of 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 and of angles A, B, and C respectively:

Circumradius, inradius, and exradius

The triangle's area is
where R is the triangle's circumradius.
We have
We also have
The ratio r /R of the inradius to the circumradius is the positive solution of the cubic equation
In addition,
We also have
In general for all integer n,
where
and
We also have
We also have
The exradius ra 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.

Trigonometric properties

The various trigonometric identities associated with the heptagonal triangle include these:
The cubic equation
has solutions and which are the squared sines of the angles of the triangle.
The positive solution of the cubic equation
equals which is twice the cosine of one of the triangle’s angles.
Sin, sin, and sin are the roots of


We also have:
For an integer n, let
For n = 0,...,20,

For n= 0, -1,,..-20,
For an integer n, let
For n= 0, 1,,..10,
For an integer n, let
For n= 0, 1,,..10,
We also have
We also have
We also have
We also have
We also have Ramanujan type identities,
We also have