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Heptagonal Triangle


HeptagonalTriangle

The unique (modulo rotations) scalene triangle formed from three vertices of a regular heptagon, having vertex angles pi/7, 2pi/7, and 4pi/7. There are a number of amazing formulas connecting the sides and angles of the heptagonal triangle (Bankoff and Garfunkel 1973).

The area of the triangle is

 A=1/4sqrt(7)R^2,
(1)

where R is the triangle's circumradius. The sum of squares of sides of the heptagonal triangle is equal to 7R^2 (Bankoff and Garfunkel 1973). The ratio x=r/R of inradius r to circumradius R is given by the positive root of

 8x^3+28x^2+14x-7=0.
(2)

The side lengths satisfy

 1/(a^2)+1/(b^2)+1/(c^2)=2/(R^2)
(3)

(Bankoff and Garfunkel 1973) and

 1/b+1/c=1/a.
(4)

The latter can be easily proved by applying Ptolemy's theorem to the quadrilateral with sides c, a, a, and b, and diagonals c and b, and dividing by abc (I. Larrosa Cañestro, pers. comm., Apr. 23, 2006).

The Brocard angle Omega satisfies

 cotOmega=sqrt(7),
(5)

and the exradius r_a is equal to the radius of the nine-point circle of DeltaABC.

a is half the harmonic mean of the other two sides,

 a=(bc)/(b+c)
(6)
 b^2-a^2=ac,
(7)

and so on for all permutations of variables (Bankoff and Garfunkel 1973). Also,

 (b^2)/(a^2)+(c^2)/(b^2)+(a^2)/(c^2)=5.
(8)

If h_a, h_b, and h_c are the altitudes, then

 h_a=h_b+h_c
(9)
 h_a^2+h_b^2+h_c^2=1/2(a^2+b^2+c^2).
(10)

If A^', B^', and C^' are the feet of the altitudes, then

 BA^'·A^'C=1/4ac
(11)

and so on (Bankoff and Garfunkel 1973). The internal angle bisectors of C and B are equal to the difference of the adjacent sides and the external angle bisector of A is equal to the sum of adjacent sides.

HeptagonalTriBisectors

The triangle DeltaDEF joining the feet of the angle bisectors of the heptagonal triangle is an isosceles triangle with DF=EF.

HeptagonalTriOrthMed
HeptagonalTriNinePoint

The orthic triangle DeltaH_AH_BH_C and median triangle M_AM_BM_C are congruent and perspective. In addition, both are similar to DeltaABC, to the pedal triangle DeltaP_AP_BP_C of DeltaABC with respect to the nine-point center N, and to the triangle DeltaII_BI_C formed by the incenter I and the exterior angle bisectors I_B and I_C (Bankoff and Garfunkel 1973). The triangle DeltaIBC is also similar to these triangles.

There are also a slew of curious trigonometric identities involving the angles of the heptagonal triangle:

cosAcosBcosC=-1/8
(12)
cos^2A+cos^2B+cos^2C=5/4
(13)
cos^4A+cos^4B+cos^4C=(13)/(16)
(14)
cos^2Acos^2B+cos^2Acos^2C+cos^2Bcos^2C=3/8
(15)
cos(2A)+cos(2B)+cos(2C)=-1/2
(16)
cotA+cotB+cotC=sqrt(7)
(17)
cot^2A+cot^2B+cot^2C=5
(18)
csc^2A+csc^2B+csc^2C=8
(19)
csc^4A+csc^4B+csc^4C=32
(20)
sec^2A+sec^2B+sec^2C=24
(21)
sec^4A+sec^4B+sec^4C=416
(22)
sec(2A)+sec(2B)+sec(2C)=-4
(23)
sinAsinBsinC=1/8sqrt(7)
(24)
sin^2A+sin^2B+sin^2C=7/4
(25)
sin(2A)+sin(2B)+sin(2C)=1/2sqrt(7)
(26)
sin^2Asin^2B+sin^2Asin^2C+sin^2Bsin^2C=7/8
(27)
sin^4A+sin^4B+sin^4C=(21)/(16)
(28)
tanAtanBtanC=-sqrt(7)
(29)
tan^2A+tan^2B+tan^2C=21
(30)

(Bankoff and Garfunkel 1973).

In addition,

 sinA+sinB+sinC=1/2cot(1/(14)pi).
(31)

Finally, the heptagonal triangle satisfies the miscellaneous properties:

1. The first Brocard point corresponds to the nine-point center and the second Brocard point lies on the nine-point circle.

2. OH=Rsqrt(2), where O is the circumcenter, H is the orthocenter, and R is the circumradius.

3. IH=(R^2+4r^2)/2, where I is the incenter and r is the inradius.

4. The two tangents from the orthocenter H to the circumcircle of the heptagonal triangle are mutually perpendicular.

5. The center of the circumcircle of the tangential triangle corresponds with the symmetric point of O with respect to H.

6. The altitude from B is half the length of the internal bisector of the angle A.


See also

Heptagon, Trigonometry Angles--Pi/7

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References

Bankoff, L. and Garfunkel, J. "The Heptagonal Triangle." Math. Mag. 46, 7-19, 1973.

Referenced on Wolfram|Alpha

Heptagonal Triangle

Cite this as:

Weisstein, Eric W. "Heptagonal Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HeptagonalTriangle.html

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