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Geometry, Vol. 1, Pages 3-15: Unary Operations on Homogeneous Coordinates in the Plane of a Triangle
Geometry doi: 10.3390/geometry1010002
Authors: Peter J. C. Moses Clark Kimberling
Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear) x:y:z. Eight unary operations discussed in this paper include u1(X)=(y−z)/x:(z−x)/y:(x−y)/z. For each ui, there exist, formally, two points, P and U, such that ui(P)=ui(U)=X. To such pairs of inverses are applied nine binary operations, each resulting in a triangle center. If L is a line, then formally, ui(L) is a cubic curve that passes through the vertices A,B,C. If L passes through the point 1:1:1 (the centroid or incenter, assuming that the coordinates are barycentric or trilinear), then the cubic is degenerate as the union of a parabola and the line at infinity. The methods in this work are largely algebraic and computer-dependent.
