coordinates of the first Brocard point are c/b : a/c : b/a. These coordinates satisfy the properties of homogeneity and cyclicity but not bisymmetry. So the first Brocard point is not (in general) a triangle center. The second Brocard point has trilinear coordinates b/c : c/a : a/b and similar remarks apply. The first and second Brocard points are one of many bicentric pairs of points, pairs of points defined from a triangle with the property that the pair (but not each individual point) is preserved under similarities of the triangle. Several binary operations, such as midpoint and trilinear product, when applied to the two Brocard points, as well as other bicentric pairs, produce triangle centers. Some well-known triangle centers Classical triangle centers Recent triangle centers In the following table of more recent triangle centers, no specific notations are mentioned for the various points. Also for each center only the first trilinear coordinate f(a,b,c) is specified. The other coordinates can be easily derived using the cyclicity property of trilinear coordinates. General classes of triangle centers Kimberling center In honor of Clark Kimberling who created the online encyclopedia of more than 32,000 triangle centers, the triangle centers listed in the encyclopedia are collectively called Kimberling centers. Polynomial triangle center A triangle center P is called a polynomial triangle center if the trilinear coordinates of P can be expressed as polynomials in a, b and c. Regular triangle center A triangle center P is called a regular triangle point if the trilinear coordinates of P can be expressed as polynomials in Δ, a, b and c, where Δ is the area of the triangle. Major triangle center A triangle center P is said to be a major triangle center if the trilinear coordinates of P can be expressed in the form f(A) : f(B) : f(C) where f(X) is a function of the angle X alone and does not depend on the other angles or on the side lengths. Transcendental triangle center A triangle center P is called a transcendental triangle center if P has no trilinear representation using only algebraic functions of a, b and c. Miscellaneous Isosceles and equilateral triangles Let f be a triangle center function. If two sides of a triangle are equal (say a = b) then so two components of the associated triangle center are always equal. Therefore, all triangle centers of an isosceles triangle must