f(b,c,a) : f(c,a,b) is called a triangle center. This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by cyclic permutation of a, b, c. This process is known as cyclicity. Every triangle center function corresponds to a unique triangle center. This correspondence is not bijective. Different functions may define the same triangle center. For example, the functions f1(a,b,c) = 1/a and f2(a,b,c) = bc both correspond to the centroid. Two triangle center functions define the same triangle center if and only if their ratio is a function symmetric in a, b and c. Even if a triangle center function is well-defined everywhere the same cannot always be said for its associated triangle center. For example, let f(a, b, c) be 0 if a/b and a/c are both rational and 1 otherwise. Then for any triangle with integer sides the associated triangle center evaluates to 0:0:0 which is undefined. Default domain In some cases these functions are not defined on the whole of ℝ3. For example, the trilinears of X365 which is the 365th entry in the Encyclopedia of Triangle Centers, are a1/2 : b1/2 : c1/2 so a, b, c cannot be negative. Furthermore, in order to represent the sides of a triangle they must satisfy the triangle inequality. So, in practice, every function's domain is restricted to the region of ℝ3 where a ≤ b + c, b ≤ c + a, and c ≤ a + b. This region T is the domain of all triangles, and it is the default domain for all triangle-based functions. Other useful domains There are various instances where it may be desirable to restrict the analysis to a smaller domain than T. For example: The centers X3, X4, X22, X24, X40 make specific reference to acute triangles, namely that region of T where a2 ≤ b2 + c2, b2 ≤ c2 + a2, c2 ≤ a2 + b2. When differentiating between the Fermat point and X13 the domain of triangles with an angle exceeding 2π/3 is important, in other words triangles for which a2 > b2 + bc + c2 or b2 > c2 + ca + a2 or c2 > a2 + ab + b2. A domain of much practical value since it