If we take the plane as B(k) : x + y + z = k, k ≠0, we describe the coordinates as the barycentric coordinates of a point on the plane. When k = 1, these are normalized barycentric coodinates. Here, the line at infinity has equation x + y + z = 0.
If we take the plane as T(k) : ax + by + cz = 0, k≠0, where a, b and c are the lengths of the sides BC, CA, AB respectively. Now, we have trilinear coordinates, and if k = 1, normalized trilinear cooordinates. In this case, the line at infinity ax + by + cz = 0. The trilinear coordinates are proportional to the signed distances of a point from the lines BC,CA,AB, respectively. The sign is chosen so that A is on the positive side of BC, and so on.
In most contexts, it is not necessary to choose a consistent k, so that only the ratios of the coordinates are relevant. To emphasise this, we write trilinear or barycentric coordinates of the projective point [u,v,w] as u:v:w.
Note that, if a point has trilinear coordinates u:v:w, then it has barycentric coordinates au:bv:cw. This is equivalent to a projective transformation preserving the respective lines at infinity. Thus, the nature of a conic as an ellipse, parabola or hyerbola is independent of the system of coordinates employed. Also, in either system, the coordinates of A,B,C are 1:0:0, 0:1:0, 0:0:1, respectively.
Definition A point P with trilinear coordinates u:v:w is a triangle
centre if there is a homogeneous function f(x,y,z) such that
Of course, for barycentric coordinates, we then have the function F(x,y,z) = xf(x,y,z).
Sometimes, to make the presentation easier, we may describe f using the angles. This works since the trigonometric functions of the angles cann be defined in terms of a,b,c. For example, in trilinear cooordinates, the circumcentre is described as the point cosA:cosB:cosC. Now, cos A = (b2+c2-a2)/2bc, so this qualifies as a triangle centre under our definition.