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
Extouch triangle Extouch triangle In geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle. The vertices of the extouch triangle are given in trilinear coordinates by: or equivalently, where "a,b,c" are the lengths of the sides opposite angles "A, B, C" respectively, The triangle's splitters are lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet at the Nagel point. This is shown in blue and labelled "N" in the diagram. The Mandart inellipse is tangent to