The unique (modulo rotations) scalene triangle formed from three vertices of a regular heptagon, having
vertex angles 
,
,
and 
.
There are a number of amazing formulas connecting the sides and angles of the heptagonal
triangle (Bankoff and Garfunkel 1973).
 |
(1)
|
where 
is the triangle's circumradius. The sum of squares
of sides of the heptagonal triangle is equal to 
(Bankoff and Garfunkel 1973). The ratio 
of inradius 
to circumradius 
is given by the positive root of
 |
(2)
|
The side lengths satisfy
 |
(3)
|
(Bankoff and Garfunkel 1973) and
 |
(4)
|
The latter can be easily proved by applying Ptolemy's theorem to the quadrilateral with sides 
, 
, 
, and 
, and diagonals 
and 
, and dividing by 
(I. Larrosa Cañestro, pers. comm., Apr. 23,
2006).
The Brocard angle 
satisfies
 |
(5)
|
and the exradius 
is equal to the radius of the nine-point
circle of 
.
is half the harmonic mean of the other two sides,
 |
(6)
|
 |
(7)
|
and so on for all permutations of variables (Bankoff and Garfunkel 1973). Also,
 |
(8)
|
If 
,
,
and 
are the altitudes, then
 |
(9)
|
 |
(10)
|
If 
,
,
and 
are the feet of the altitudes, then
 |
(11)
|
and so on (Bankoff and Garfunkel 1973). The internal angle bisectors of 
and 
are equal to the difference of the adjacent sides and the
external angle bisector of 
is equal to the sum of adjacent sides.
The triangle 
joining the feet of the angle bisectors of the heptagonal triangle is an isosceles
triangle with 
.
 |  |
The orthic triangle 
and median triangle 
are congruent and perspective.
In addition, both are similar to 
, to the pedal triangle 
of 
with respect to the nine-point
center 
,
and to the triangle 
formed by the incenter 
and the exterior angle bisectors 
and 
(Bankoff and Garfunkel 1973). The triangle 
is also similar to these triangles.
There are also a slew of curious trigonometric identities involving the angles of the heptagonal triangle:
 |
(12)
|
 |
(13)
|
 |
(14)
|
 |
(15)
|
 |
(16)
|
 |
(17)
|
 |
(18)
|
 |
(19)
|
 |
(20)
|
 |
(21)
|
 |
(22)
|
 |
(23)
|
 |
(24)
|
 |
(25)
|
 |
(26)
|
 |
(27)
|
 |
(28)
|
 |
(29)
|
 |
(30)
|
(Bankoff and Garfunkel 1973).
In addition,
 |
(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. 
,
where 
is the circumcenter, 
is the orthocenter, and 
is the circumradius.
3. 
,
where 
is the incenter and 
is the inradius.
4. The two tangents from the orthocenter 
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 
with respect to 
.
6. The altitude from 
is half the length of the internal bisector of the angle 
.
