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
Skinny triangle based on the small-angle approximations; and, when formula_5 is in radians. The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1; The term formula_7 represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus "θ") are equal to "π". Applying the small angle approximations to the law of sines above results in; the desired result. This result is equivalent to assuming that the length of the base of