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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 \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. 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 :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 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 : \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 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 ...
: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 have :\frac, \quad -\frac, \quad -\frac 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 ...
:t^3+4t^2+3t-1=0. We also have :\frac, \quad -\frac, \quad \frac 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 ...
:t^3-t^2-9t+1=0. We also have :\frac, \quad \frac, \quad -\frac 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 ...
: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.


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 have :a^4+b^4+c^4=21R^4. :a^6+b^6+c^6=70R^6. 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 :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 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 A,B,C,G=X(2),H=X(4) has the following properties: * first focus F_1 = X(5) * center U is on Euler circle (general property) and on circle (O, F_1) * second focus F_2 is on the circumcircle


Trigonometric properties


Trigonometric identities

The various trigonometric identities associated with the heptagonal triangle include these: \begin A &= \frac \\ pt \cos A &= \frac \end \quad\begin B &= \frac \\ pt \cos B &= \frac \end \quad\begin C &= \frac \\ pt \cos C &= -\frac \end \begin \sin A \!&\! \times \!&\! \sin B \!&\! \times \!&\! \sin C \!&\! = \!&\! \frac \\ pt \sin A \!&\! - \!&\! \sin B \!&\! - \!&\! \sin C \!&\! = \!&\! -\frac \\ pt \cos A \!&\! \times \!&\! \cos B \!&\! \times \!&\! \cos C \!&\! = \!&\! -\frac \\ pt \tan A \!&\! \times \!&\! \tan B \!&\! \times \!&\! \tan C \!&\! = \!&\! -\sqrt \\ pt \tan A \!&\! + \!&\! \tan B \!&\! + \!&\! \tan C \!&\! = \!&\! -\sqrt \\ pt \cot A \!&\! + \!&\! \cot B \!&\! + \!&\! \cot C \!&\! = \!&\! \sqrt \\ pt \sin^2\!A \!&\! \times \!&\! \sin^2\!B \!&\! \times \!&\! \sin^2\!C \!&\! = \!&\! \frac \\ pt \sin^2\!A \!&\! + \!&\! \sin^2\!B \!&\! + \!&\! \sin^2\!C \!&\! = \!&\! \frac \\ pt \cos^2\!A \!&\! + \!&\! \cos^2\!B \!&\! + \!&\! \cos^2\!C \!&\! = \!&\! \frac \\ pt \tan^2\!A \!&\! + \!&\! \tan^2\!B \!&\! + \!&\! \tan^2\!C \!&\! = \!&\! 21 \\ pt \sec^2\!A \!&\! + \!&\! \sec^2\!B \!&\! + \!&\! \sec^2\!C \!&\! = \!&\! 24 \\ pt \csc^2\!A \!&\! + \!&\! \csc^2\!B \!&\! + \!&\! \csc^2\!C \!&\! = \!&\! 8 \\ pt \cot^2\!A \!&\! + \!&\! \cot^2\!B \!&\! + \!&\! \cot^2\!C \!&\! = \!&\! 5 \\ pt \sin^4\!A \!&\! + \!&\! \sin^4\!B \!&\! + \!&\! \sin^4\!C \!&\! = \!&\! \frac \\ pt \cos^4\!A \!&\! + \!&\! \cos^4\!B \!&\! + \!&\! \cos^4\!C \!&\! = \!&\! \frac \\ pt \sec^4\!A \!&\! + \!&\! \sec^4\!B \!&\! + \!&\! \sec^4\!C \!&\! = \!&\! 416 \\ pt \csc^4\!A \!&\! + \!&\! \csc^4\!B \!&\! + \!&\! \csc^4\!C \!&\! = \!&\! 32 \\ pt\end \begin \tan A \!&\! - \!&\! 4\sin B \!&\! = \!&\! -\sqrt \\ pt \tan B \!&\! - \!&\! 4\sin C \!&\! = \!&\! -\sqrt \\ pt \tan C \!&\! + \!&\! 4\sin A \!&\! = \!&\! -\sqrt \end \begin \cot^2\! A &= 1 -\frac \\ pt \cot^2\! B &= 1 -\frac \\ pt \cot^2\! C &= 1 -\frac \end \begin \cos A \!&\! = \!&\! \frac \!&\! + \!&\! \frac \!&\! \times \!&\! \sin^3\! C \\ pt \sec A \!&\! = \!&\! 2 \!&\! + \!&\! 4 \!&\! \times \!&\! \cos C \\ pt \sec A \!&\! = \!&\! 6 \!&\! - \!&\! 8 \!&\! \times \!&\! \sin^2\! B \\ pt \sec A \!&\! = \!&\! 4 \!&\! - \!&\! \frac \!&\! \times \!&\! \sin^3\! B \\ pt \cot A \!&\! = \!&\! \sqrt \!&\! + \!&\! \frac \!&\! \times \!&\! \sin^2\! B \\ pt \cot A \!&\! = \!&\! \frac \!&\! + \!&\! \frac \!&\! \times \!&\! \cos B \\ pt \sin^2\! A \!&\! = \!&\! \frac \!&\! + \!&\! \frac \!&\! \times \!&\! \cos B \\ pt \cos^2\! A \!&\! = \!&\! \frac \!&\! + \!&\! \frac \!&\! \times \!&\! \sin^3\! A \\ pt \cot^2\! A \!&\! = \!&\! 3 \!&\! + \!&\! \frac \!&\! \times \!&\! \sin A \\ pt \sin^3\! A \!&\! = \!&\! \frac \!&\! + \!&\! \frac \!&\! \times \!&\! \cos B \\ pt \csc^3\! A \!&\! = \!&\! \frac \!&\! + \!&\! \frac \!&\! \times \!&\! \tan^2\! C \end \sin A\sin B - \sin B\sin C + \sin C\sin A = 0 \begin \sin^3\!B\sin C - \sin^3\!C\sin A - \sin^3\!A\sin B &= 0 \\ pt\sin B\sin^3\!C - \sin C\sin^3\!A - \sin A\sin^3\!B &= \frac \\ pt\sin^4\!B\sin C - \sin^4\!C\sin A + \sin^4\!A\sin B &= 0 \\ pt\sin B\sin^4\!C + \sin C\sin^4\!A - \sin A\sin^4\!B &= \frac \end \begin \sin^\!B\sin^3\!C - \sin^\!C\sin^3\!A - \sin^\!A\sin^3\!B &= 0 \\ pt\sin^3\!B\sin^\!C - \sin^3\!C\sin^\!A - \sin^3\!A\sin^\!B &= \frac \end


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 ...
64y^3-112y^2+56y-7=0 has solutions \sin^2\! A,\ \sin^2\! B,\ \sin^2\! C. The positive solution of the cubic equation x^3+x^2-2x-1=0 equals 2\cos B. 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 x^3 - \tfracx^2 + \tfrac = 0 are \sin 2A,\ \sin 2B,\ \sin 2C. The roots of the cubic equation x^3 - \tfrac x^2 + \tfrac = 0 are -\sin A,\ \sin B,\ \sin C. The roots of the cubic equation x^3 + \tfracx^2 - \tfracx - \tfrac = 0 are -\cos A,\ \cos B,\ \cos C. The roots of the cubic equation x^3 + \sqrtx^2 - 7x + \sqrt = 0 are \tan A,\ \tan B,\ \tan C. The roots of the cubic equation x^3 - 21x^2 + 35x - 7 = 0 are \tan^2\! A,\ \tan^2\! B,\ \tan^2\! C.


Sequences

For an integer , let \begin S(n) &= (-\sin A)^n + \sin^n\! B + \sin^n\! C \\ ptC(n) &= (-\cos A)^n + \cos^n\! B + \cos^n\! C \\ ptT(n) &= \tan^n\! A + \tan^n\! B + \tan^n\! C \end


Ramanujan identities

We also have Ramanujan type identities, \begin \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! -\sqrt 8\times \sqrt \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! -\sqrt 8\times \sqrt \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! \sqrt 8\times \sqrt \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! \sqrt \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! \sqrt \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! -\sqrt 8\times \sqrt \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! \sqrt 8\times \sqrt \end \begin \frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! -\frac \times \sqrt \\ pt \frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \frac \times \sqrt \\ pt \frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \sqrt \\ pt \frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \sqrt \\ pt \frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! -\frac \times \sqrt \\ pt \frac \!&\! + \!&\! \frac \!&\! + \!&\! \frac \!&\! = \!&\! \frac \times \sqrt \end \begin \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! -\sqrt \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! 0 \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! -\frac \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! 0 \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! -3\times \frac \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! 0 \\ pt \sqrt \!&\! + \!&\! \sqrt \!&\! + \!&\! \sqrt \!&\! = \!&\! -61\times \frac. \end


References

{{reflist Types of triangles