Obtuse triangle formed by the side and diagonals of a regular heptagon
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 formula_1 and formula_2 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
formula_3
where "R" is the circumradius. The squared distance from the incenter "I" to the orthocenter is
formula_4
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
formula_5
(the latter being the optic equation) and hence
formula_6
and
formula_7
formula_8
formula_9
Thus –"b"/"c", "c"/"a", and "a"/"b" all satisfy the cubic equation
formula_10
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
formula_11
We also have
formula_12 
satisfy the cubic equation
formula_13
We also have
formula_14 
satisfy the cubic equation
formula_15
We also have
formula_16 
satisfy the cubic equation
formula_17
We also have
formula_18
formula_19
formula_20
and
formula_21
We also have
formula_22 
formula_23 
formula_24 
formula_25
Altitudes.
The altitudes "h""a", "h""b", and "h""c" satisfy
formula_26
and
formula_27
The altitude from side "b" (opposite angle "B") is half the internal angle bisector formula_28 of "A":
formula_29
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 formula_30 and formula_31 of angles "A, B", and "C" respectively:
formula_32
formula_33
formula_34
Circumradius, inradius, and exradius.
The triangle's area is
formula_35
where "R" is the triangle's circumradius.
We have
formula_36
We also have
formula_37
formula_38
The ratio "r" /"R" of the inradius to the circumradius is the positive solution of the cubic equation
formula_39
In addition,
formula_40
We also have
formula_41
formula_42
In general for all integer "n", 
formula_43
where 
formula_44
and 
formula_45
We also have
formula_46
We also have
formula_47 
formula_48 
formula_49
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).
Hyperbola.
The rectangular hyperbola through formula_50 has the following properties:
Trigonometric properties.
Trigonometric identities.
The various trigonometric identities associated with the heptagonal triangle include these:
formula_55
formula_56
formula_57
formula_58
formula_59
formula_60
formula_61 
formula_62
Cubic polynomials.
The cubic equation formula_63 has solutions formula_64
The positive solution of the cubic equation formula_65 equals formula_66
The roots of the cubic equation formula_67 are formula_68
The roots of the cubic equation formula_69 are formula_70
The roots of the cubic equation formula_71 are formula_72
The roots of the cubic equation formula_73 are formula_74
The roots of the cubic equation formula_75 are formula_76
Sequences.
For an integer n, let 
formula_77
Ramanujan identities.
We also have Ramanujan type identities,
formula_78
formula_79
formula_80
formula_81
References.
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