Trigonometric Polynomial Points in the Plane of a Triangle

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths a,b, and c of a triangle ABC. For example, the circumcenter is represented by the polynomial a(b2+c2−a2). It is not so well known that triangle centers have barycentric coordinates, such as tanA : tan B : tan C, that are also representable by polynomials, in this case, by p(a, b, c) : p(b, c, a) : p(c, a, b), where p(a, b, c)=a(a2+b2−c2)(a2+c2−b2). This paper presents and discusses the polynomial representations of triangle centers that have barycentric coordinates of the form f(a, b, c) : f(b, c, a) : f(c, a, b), where f depends on one or more of the functions in the set {cos, sin, tan, sec, csc, cot}. The topics discussed include infinite trigonometric orthopoints, the n-Euler line, and symbolic substitution.