A point "E" inside a triangle "ABC" can be coordinatized by the areas of the triangles "EBC," "ECA," and "EAB." These are called the barycentric coordinates of "E." It can also be coordinatized using the six segments into which the cevians through "E" divide the sides of "ABC," or the six angles into which the cevians through "E" divide the angles of "ABC," or the six triangles into which the cevians through "E" divide "ABC," etc. This article introduces several coordinate systems of these types, and investigates those centres of "ABC" whose coordinates, relative to a given coordinate system, are linear (or quasi-linear) with respect to appropriate elements of "ABC," such as its side-lengths, its angles, etc. This results in grouping known centres into new families, and in discovering new centres. It also leads to unifying several results that are scattered in the literature, and creates several open questions that may be suitable for classroom discussions and team projects in which algebra and geometry packages are expected to be useful. These questions may also be used for Mathematical Olympiad training and may serve as supplementary material for students taking a course in Euclidean geometry. (Contains 10 figures.)