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)
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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)
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The side lengths satisfy
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(Bankoff and Garfunkel 1973) and
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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
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and the exradius is equal to the radius of the nine-point circle of .
is half the harmonic mean of the other two sides,
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and so on for all permutations of variables (Bankoff and Garfunkel 1973). Also,
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If , , and are the altitudes, then
(9)
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(10)
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If , , and are the feet of the altitudes, then
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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 .
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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)
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(13)
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(18)
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(21)
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(22)
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(23)
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(27)
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(29)
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(30)
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(Bankoff and Garfunkel 1973).
In addition,
(31)
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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 .