Given a circle expressed in trilinear coordinates by (l α + m β + n γ)(a α + b β + c γ) + k(a βγ + b γα + c αβ) = 0, a central circle is a circle such that l:m:n is a triangle center and k is a homogeneous function that is symmetric in the side lengths a, b, and c. The following table summarizes the triangle centers whose trilinears correspond to a circle with l:m:n (for some appropriate value of k). In the table, * indicated a circle function that is known but which does not appear among the list of Kimberling centers. Note also that the circumcircle is not actually a central circle, since its trilinears 0:0:0 are not those of a triangle center.

central conic | central line | circle | circle function