Heptagonal triangle 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). The various trigonometric identities associated with the heptagonal triangle include these:, The cubic equation has solutions formula_66 and formula_67 which are the squared sines of the angles of the triangle. The positive solution of the cubic equation equals formula_69 which is twice the cosine of one of the triangle’s angles. We also have: For an
Schwarz triangle function of the upper half plane. To see how to handle the case when an angle of Δ is a right angle, note that the inequality implies that if one of the angles is a right angle, say "a" = 2, then both "b" and "c" are greater than 2 and one of them, "b" say, must be greater than 3. In this case, reflecting the triangle across the side AB gives an isosceles hyperbolic triangle with angles /"c", /"c" and 2/"b". If 2/"b" ≤ /3, i.e. "b" is greater than 5, then all the angles of the doubled triangle are