Overview
heptagonal triangles has one green side, one blue side, and one red side. 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 \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. The distance between the circumcenter O and the orthocenter H is given by : OH=R\sqrt 2 , where R is the circumradius. The squared distance from the incenter I to the orthocenter is : IH^2=\frac R^2+4r^2 2 , where r is the inradius. The two tangents from the orthocenter to the circumcircle are mutually perpendicular.
Sides
The heptagonal triangle's sides a \begin align a^2 & =c(c-b), \\ 5pt b^2 & =a(c+a), \\ 5pt c^2 & =b(a+b), \\ 5pt \frac 1 a & =\frac 1 b + \frac 1 c \end align (the latter : \frac a^2 bc , \quad -\frac b^2 ca , \quad -\frac c^2 ab satisfy the cubic equation : t^3+4t^2+3t-1=0. We also have : \frac a^3 bc^2 , \quad -\frac b^3 ca^2 , \quad \frac c^3 ab^2 satisfy the cubic equation : t^3-t^2-9t+1=0. We also have : \frac a^3 b^2c , \quad \frac b^3 c^2a , \quad -\frac c^3 a^2b 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 b^2 a^2 +\frac c^2 b^2 +\frac a^2 c^2 =5. We also have : ab-bc+ca=0, : a^ 3 b-b^ 3 c+c^ 3 a=0, : a^ 4 b+b^ 4 c-c^ 4 a=0, : a^ 11 b^ 3 -b^ 11 c^ 3 +c^ 11 a^ 3 =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 a^2+b^2+c^2 2 . 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^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 1 a^2 +\frac 1 b^2 +\frac 1 c^2 =\frac 2 R^2 . We also have : \frac 1 a^4 +\frac 1 b^4 +\frac 1 c^4 =\frac 2 R^4 . : \frac 1 a^6 +\frac 1 b^6 +\frac 1 c^6 =\frac 17 7R^6 . In general for all integer n, : a^ 2n +b^ 2n +c^ 2n =g(n)(2R)^ 2n 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=\sqrt 7 bR, \quad 2c^2-b^2=\sqrt 7 cR, \quad 2a^2-c^2=-\sqrt 7 aR. We also have : a^ 3 c + b^ 3 a - c^ 3 b = -7R^ 4 , : a^ 4 c - b^ 4 a + c^ 4 b = 7\sqrt 7 R^ 5 , : a^ 11 c^ 3 +b^ 11 a^ 3 - c^ 11 b^ 3 = -7^ 3 17R^ 14 . 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 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 identities
The various trigonometric identities associated with the heptagonal triangle include these: \begin align \cot^2\! A &= 1 -\frac 2 \tan C \sqrt 7 \\ 2pt \cot^2\! B &= 1 -\frac 2 \tan A \sqrt 7 \\ 2pt \cot^2\! C &= 1 -\frac 2 \tan B \sqrt 7 \end align
Cubic polynomials
The cubic equation 64y^3-112y^2+56y-7=0 has solutions The roots of the cubic equation x^3 - \tfrac \sqrt 7 2 x^2 + \tfrac \sqrt 7 8 = 0 are \sin 2A,\ \sin 2B,\ \sin 2C. The roots of the cubic equation x^3 - \tfrac \sqrt 7 2 x^2 + \tfrac \sqrt 7 8 = 0 are -\sin A,\ \sin B,\ \sin C. The roots of the cubic equation x^3 + \tfrac 1 2 x^2 - \tfrac 1 2 x - \tfrac 1 8 = 0 are -\cos A,\ \cos B,\ \cos C. The roots of the cubic equation x^3 + \sqrt 7 x^2 - 7x + \sqrt 7 = 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 align S(n) &= (-\sin A)^n + \sin^n\! B + \sin^n\! C \\ 4pt C(n) &= (-\cos A)^n + \cos^n\! B + \cos^n\! C \\ 4pt T(n) &= \tan^n\! A + \tan^n\! B + \tan^n\! C \end align
Ramanujan identities
We also have Ramanujan type identities, \begin array ccccccl \sqrt 3 2\sin 2A \!&\! + \!&\! \sqrt 3 2\sin 2B \!&\! + \!&\! \sqrt 3 2\sin 2C \!&\! = \!&\! -\sqrt 18 7 \times \sqrt 3 -\sqrt 3 7 + 6 + 3\left(\sqrt 3 5 - 3 \sqrt 3 7 + \sqrt 3 4 - 3 \sqrt 3 7 \right) \\ 2pt \sqrt 3 2\sin 2A \!&\! + \!&\! \sqrt 3 2\sin 2B \!&\! + \!&\! \sqrt 3 2\sin 2C \!&\! = \!&\! -\sqrt 18 7 \times \sqrt 3 -\sqrt 3 7 + 6 + 3\left(\sqrt 3 5 - 3 \sqrt 3 7 + \sqrt 3 4 - 3 \sqrt 3 7 \right) \\ 2pt \sqrt 3 4\sin^2 2A \!&\! + \!&\! \sqrt 3 4\sin^2 2B \!&\! + \!&\! \sqrt 3 4\sin^2 2C \!&\! = \!&\! \sqrt 18 49 \times \sqrt 3 \sqrt 3 49 + 6 + 3\left(\sqrt 3 12 + 3( \sqrt 3 49 + 2\sqrt 3 7 ) + \sqrt 3 11 + 3( \sqrt 3 49 + 2\sqrt 3 7 ) \right) \\ 6pt \sqrt 3 2\cos 2A \!&\! + \!&\! \sqrt 3 2\cos 2B \!&\! + \!&\! \sqrt 3 2\cos 2C \!&\! = \!&\! \sqrt 3 5 - 3\sqrt 3 7 \\ 8pt \sqrt 3 4\cos^2 2A \!&\! + \!&\! \sqrt 3 4\cos^2 2B \!&\! + \!&\! \sqrt 3 4\cos^2 2C \!&\! = \!&\! \sqrt 3 11 + 3(2\sqrt 3 7 + \sqrt 3 49 ) \\ 6pt \sqrt 3 \tan 2A \!&\! + \!&\! \sqrt 3 \tan 2B \!&\! + \!&\! \sqrt 3 \tan 2C \!&\! = \!&\! -\sqrt 18 7 \times \sqrt 3 \sqrt 3 7 + 6 + 3\left(\sqrt 3 5 + 3(\sqrt 3 7 - \sqrt 3 49 ) + \sqrt 3 - 3 + 3(\sqrt 3 7 - \sqrt 3 49 ) \right) \\ 2pt \sqrt 3 \tan^2 2A \!&\! + \!&\! \sqrt 3 \tan^2 2B \!&\! + \!&\! \sqrt 3 \tan^2 2C \!&\! = \!&\! \sqrt 18 49 \times \sqrt 3 3\sqrt 3 49 + 6 + 3\left(\sqrt 3 89 + 3(3\sqrt 3 49 + 5\sqrt 3 7 ) + \sqrt 3 25 + 3(3\sqrt 3 49 + 5\sqrt 3 7 ) \right) \end array \begin array ccccccl \frac 1 \sqrt 3 2\sin 2A \!&\! + \!&\! \frac 1 \sqrt 3 2\sin 2B \!&\! + \!&\! \frac 1 \sqrt 3 2\sin 2C \!&\! = \!&\! -\frac 1 \sqrt 18 7 \times \sqrt 3 6 + 3\left(\sqrt 3 5 - 3 \sqrt 3 7 + \sqrt 3 4 - 3 \sqrt 3 7 \right) \\ 2pt \frac 1 \sqrt 3 4\sin^2 2A \!&\! + \!&\! \frac 1 \sqrt 3 4\sin^2 2B \!&\! + \!&\! \frac 1 \sqrt 3 4\sin^2 2C \!&\! = \!&\! \frac 1 \sqrt 18 49 \times \sqrt 3 2\sqrt 3 7 + 6 + 3\left(\sqrt 3 12 + 3( \sqrt 3 49 + 2\sqrt 3 7 ) + \sqrt 3 11 + 3( \sqrt 3 49 + 2\sqrt 3 7 ) \right) \\ 2pt \frac 1 \sqrt 3 2\cos 2A \!&\! + \!&\! \frac 1 \sqrt 3 2\cos 2B \!&\! + \!&\! \frac 1 \sqrt 3 2\cos 2C \!&\! = \!&\! \sqrt 3 4 - 3\sqrt 3 7 \\ 6pt \frac 1 \sqrt 3 4\cos^2 2A \!&\! + \!&\! \frac 1 \sqrt 3 4\cos^2 2B \!&\! + \!&\! \frac 1 \sqrt 3 4\cos^2 2C \!&\! = \!&\! \sqrt 3 12 + 3(2\sqrt 3 7 + \sqrt 3 49 ) \\ 2pt \frac 1 \sqrt 3 \tan 2A \!&\! + \!&\! \frac 1 \sqrt 3 \tan 2B \!&\! + \!&\! \frac 1 \sqrt 3 \tan 2C \!&\! = \!&\! -\frac 1 \sqrt 18 7 \times \sqrt 3 -\sqrt 3 49 + 6 + 3\left(\sqrt 3 5 + 3(\sqrt 3 7 - \sqrt 3 49 ) + \sqrt 3 - 3 + 3(\sqrt 3 7 - \sqrt 3 49 ) \right) \\ 2pt \frac 1 \sqrt 3 \tan^2 2A \!&\! + \!&\! \frac 1 \sqrt 3 \tan^2 2B \!&\! + \!&\! \frac 1 \sqrt 3 \tan^2 2C \!&\! = \!&\! \frac 1 \sqrt 18 49 \times \sqrt 3 5\sqrt 3 7 + 6 + 3\left(\sqrt 3 89 + 3(3\sqrt 3 49 + 5\sqrt 3 7 ) + \sqrt 3 25 + 3(3\sqrt 3 49 + 5\sqrt 3 7 ) \right) \end array \begin array ccccccl \sqrt 3 \frac \cos 2A \cos 2B \!&\! + \!&\! \sqrt 3 \frac \cos 2B \cos 2C \!&\! + \!&\! \sqrt 3 \frac \cos 2C \cos 2A \!&\! = \!&\! -\sqrt 3 7 \\ 2pt \sqrt 3 \frac \cos 2B \cos 2A \!&\! + \!&\! \sqrt 3 \frac \cos 2C \cos 2B \!&\! + \!&\! \sqrt 3 \frac \cos 2A \cos 2C \!&\! = \!&\!
0 \\ 2pt \sqrt 3 \frac \cos^4 2B \cos 2A \!&\! + \!&\! \sqrt 3 \frac \cos^4 2C \cos 2B \!&\! + \!&\! \sqrt 3 \frac \cos^4 2A \cos 2C \!&\! = \!&\! -\frac \sqrt 3 49 2 \\ 2pt \sqrt 3 \frac \cos^5 2A \cos^2 2B \!&\! + \!&\! \sqrt 3 \frac \cos^5 2B \cos^2 2C \!&\! + \!&\! \sqrt 3 \frac \cos^5 2C \cos^2 2A \!&\! = \!&\! 0 \\ 2pt \sqrt 3 \frac \cos^5 2B \cos^2 2A \!&\! + \!&\! \sqrt 3 \frac \cos^5 2C \cos^2 2B \!&\! + \!&\! \sqrt 3 \frac \cos^5 2A \cos^2 2C \!&\! = \!&\! -3\times \frac \sqrt 3 7 2 \\ 2pt \sqrt 3 \frac \cos^ 14 2A \cos^5 2B \!&\! + \!&\! \sqrt 3 \frac \cos^ 14 2B \cos^5 2C \!&\! + \!&\! \sqrt 3 \frac \cos^ 14 2C \cos^5 2A \!&\! = \!&\! 0 \\ 2pt \sqrt 3 \frac \cos^ 14 2B \cos^5 2A \!&\! + \!&\! \sqrt 3 \frac \cos^ 14 2C \cos^5 2B \!&\! + \!&\! \sqrt 3 \frac \cos^ 14 2A \cos^5 2C \!&\! = \!&\! -61\times \frac \sqrt 3 7 8 . \end array