Heptagonal triangle

A regular heptagon (with red sides), its longer diagonals (green), and its shorter diagonals (blue). Each of the fourteen heptagonal triangles has one green side, one blue side, and one red side.

A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbtrary 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 ${\displaystyle \pi /7,2\pi /7,}$ and ${\displaystyle 4\pi /7,}$ 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.^{[1]:Propos. 12}

The second Brocard point lies on the nine-point circle.^{[2]:p. 19}

The circumcenter and the Fermat points of a heptagonal triangle form an equilateral triangle.^{[1]:Thm. 22}

The distance between the circumcenter O and the orthocenter H is given by^{[2]:p. 19}

${\displaystyle OH=R{\sqrt {2}},}$

where R is the circumradius. The squared distance from the incenter I to the orthocenter is^{[2]:p. 19}

${\displaystyle IH^{2}={\frac {R^{2}+4r^{2}}{2}},}$

where r is the inradius.

The two tangents from the orthocenter to the circumcircle are mutually perpendicular.^{[2]:p. 19}

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^{[3]:Lemma 1}

${\displaystyle {\begin{aligned}a^{2}&=c(c-b),\\[5pt]b^{2}&=a(c+a),\\[5pt]c^{2}&=b(a+b),\\[5pt]{\frac {1}{a}}&={\frac {1}{b}}+{\frac {1}{c}}\end{aligned}}}$

(the latter^{[2]:p. 13} being the optic equation) and hence

${\displaystyle ab+ac=bc,}$

and^{[3]:Coro. 2}

${\displaystyle b^{3}+2b^{2}c-bc^{2}-c^{3}=0,}$

${\displaystyle c^{3}-2c^{2}a-ca^{2}+a^{3}=0,}$

${\displaystyle a^{3}-2a^{2}b-ab^{2}+b^{3}=0.}$

Thus –b/c, c/a, and a/b all satisfy the cubic equation

${\displaystyle t^{3}-2t^{2}-t+1=0.}$

However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis.

We also have^{[2]:p. 14}

${\displaystyle b^{2}-a^{2}=ac,}$

${\displaystyle c^{2}-b^{2}=ab,}$

${\displaystyle a^{2}-c^{2}=-bc,}$

and^{[2]:p. 15}

${\displaystyle {\frac {b^{2}}{a^{2}}}+{\frac {c^{2}}{b^{2}}}+{\frac {a^{2}}{c^{2}}}=5.}$

Altitudes

The altitudes h_a, h_b, and h_c satisfy

${\displaystyle h_{a}=h_{b}+h_{c}}$^{[2]:p. 13}

and

${\displaystyle h_{a}^{2}+h_{b}^{2}+h_{c}^{2}={\frac {a^{2}+b^{2}+c^{2}}{2}}.}$^{[2]:p. 14}

The altitude from side b (opposite angle B) is half the internal angle bisector ${\displaystyle w_{A}}$ of A.^{[2]:p. 19}

${\displaystyle 2h_{b}=w_{A}.}$

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 ${\displaystyle w_{A},w_{B},}$ and ${\displaystyle w_{C}}$ of angles A, B, and C respectively:^{[2]:p. 16}

${\displaystyle w_{A}=b+c,}$

${\displaystyle w_{B}=c-a,}$

${\displaystyle w_{C}=b-a.}$

Circumradius, inradius, and exradius

The triangle's area is^[4]

${\displaystyle A={\frac {\sqrt {7}}{4}}R^{2},}$

where R is the triangle's circumradius.

We have^{[2]:p. 12}

${\displaystyle a^{2}+b^{2}+c^{2}=7R^{2}.}$

The ratio r /R of the inradius to the circumradius is the positive solution of the cubic equation^[4]

${\displaystyle 8x^{3}+28x^{2}+14x-7=0.}$

In addition,^{[2]:p. 15}

${\displaystyle {\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}={\frac {2}{R^{2}}}.}$

The exradius r_a corresponding to side a equals the radius of the nine-point circle of the heptagonal triangle.^{[2]:p. 15}

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).^{[2]:pp. 12–13}

Trigonometric properties

The various trigonometric identities associated with the heptagonal triangle include these:^{[2]:pp. 13–14[4]}

${\displaystyle \cos A\cos B\cos C=-{\frac {1}{8}},}$

${\displaystyle \cos ^{2}A+\cos ^{2}B+\cos ^{2}C={\frac {5}{4}},}$

${\displaystyle \cos ^{4}A+\cos ^{4}B+\cos ^{4}C={\frac {13}{16}},}$

${\displaystyle \cot A+\cot B+\cot C={\sqrt {7}},}$

${\displaystyle \cot ^{2}A+\cot ^{2}B+\cot ^{2}C=5,}$

${\displaystyle \csc ^{2}A+\csc ^{2}B+\csc ^{2}C=8,}$

${\displaystyle \csc ^{4}A+\csc ^{4}B+\csc ^{4}C=32,}$

${\displaystyle \sec ^{2}A+\sec ^{2}B+\sec ^{2}C=24,}$

${\displaystyle \sec ^{4}A+\sec ^{4}B+\sec ^{4}C=416,}$

${\displaystyle \sin A\sin B\sin C={\frac {\sqrt {7}}{8}},}$

${\displaystyle \sin ^{2}A\sin ^{2}B\sin ^{2}C={\frac {7}{64}},}$

${\displaystyle \sin A+\sin B+\sin C={\sqrt {\frac {7}{2}}},}$

${\displaystyle \sin ^{2}A+\sin ^{2}B+\sin ^{2}C={\frac {7}{4}},}$

${\displaystyle \sin ^{4}A+\sin ^{4}B+\sin ^{4}C={\frac {21}{16}},}$

${\displaystyle \tan A\tan B\tan C=\tan A+\tan B+\tan C=-{\sqrt {7}},}$

${\displaystyle \tan ^{2}A+\tan ^{2}B+\tan ^{2}C=21.}$

The cubic equation

${\displaystyle 64y^{3}-112y^{2}+56y-7=0}$

has solutions^{[2]:p. 14} ${\displaystyle \sin ^{2}{\frac {\pi }{7}},\sin ^{2}{\frac {2\pi }{7}},}$ and ${\displaystyle \sin ^{2}{\frac {4\pi }{7}},}$ which are the squared sines of the angles of the triangle.

The positive solution of the cubic equation

${\displaystyle x^{3}+x^{2}-2x-1=0}$

equals ${\displaystyle 2\cos {\frac {2\pi }{7}}.}$^{[5]:p. 186–187}

References

1. Paul Yiu, "Heptagonal Triangles and Their Companions", Forum Geometricorum 9, 2009, 125–148. http://forumgeom.fau.edu/FG2009volume9/FG200912.pdf
2. Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", Mathematics Magazine 46 (1), January 1973, 7–19.
3. Abdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", Forum Geometricorum 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf
4. Weisstein, Eric W. "Heptagonal Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HeptagonalTriangle.html
5. Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. Archived from the original (PDF) on 2015-12-19.