In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, 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.
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
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
.
[
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
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
.
The two tangents from the orthocenter to the circumcircle are mutually perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
.[
]
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
:
(the latter[ being the optic equation) and hence
:
and][
:
:
:
Thus –''b''/''c'', ''c''/''a'', and ''a''/''b'' all satisfy the ]cubic equation
In algebra, a cubic equation in one variable is an equation of the form
ax^3+bx^2+cx+d=0
in which is not zero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
:
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
In algebra, a cubic equation in one variable is an equation of the form
ax^3+bx^2+cx+d=0
in which is not zero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
:
We also have[
:
satisfy the ]cubic equation
In algebra, a cubic equation in one variable is an equation of the form
ax^3+bx^2+cx+d=0
in which is not zero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
:
We also have[
:
satisfy the ]cubic equation
In algebra, a cubic equation in one variable is an equation of the form
ax^3+bx^2+cx+d=0
in which is not zero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
:
We also have[
:
:
:
and][
:
We also have][
:
:
:
:
]
Altitudes
The altitudes ''h''''a'', ''h''''b'', and ''h''''c'' satisfy
:[
and
:][
The altitude from side ''b'' (opposite angle ''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
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
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 ''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
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
being the only acute one).[
]
Hyperbola
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
Trigonometric identities
The various trigonometric identities associated with the heptagonal triangle include these:[
][
]
[
][
]
Cubic polynomials
The cubic equation
In algebra, a cubic equation in one variable is an equation of the form
ax^3+bx^2+cx+d=0
in which is not zero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
has solutions[
The positive solution of the cubic equation equals ]
The roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
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
The roots of the cubic equation are
]
Sequences
For an integer , let
Ramanujan identities
We also have Ramanujan type identities,
[
]
References
{{reflist
Types of triangles