There are many points, associated with a triangle. Familiar examples are the incentre, centroid
and circumcentre. At the end of 2003, there were no fewer than 2445 such points. The definitive
list is the Encyclopedia of Triangle Centres maintained by Clark Kimberling.
We shall adopt
Kimberling's notation, so the three centres cited above are X(1), X(2) and X(3). We shall use
barycentric coordinates in general, so that the incentre X(1) is a:b:c, where a,b,c are the lengths
of the sides of the triangle.
We refer the reader to Kimberling's list for the definitions of centres.
Here, we have some pages relating to the geometry and algebra of centres.