This report will be presenting a generalization to a previous method to solveEuclidean geometry problems which are parameterizable in one variable, known asthe Method of Moving Points". This method sometimes faces limitations, oftenunable to directly intersect or parameterize curves with degrees greater than onewithout tailored geometric analysis. We generalize this method through applying theVeronese map to be able to parameterize higher-degree moving curves, and extendthe notion of multiplicity of point-point concurrence to the degree of vanishing of adeterminant, to find effective bounds on the degree of higher-degree moving curves.Additionally, through an application of polynomial resultants, we bound the degreeof the locus of intersections of higher-degree moving curves. Finally, we present acollection of examples and applications of this theory to solving olympiad geometry problems involving moving circles and factoring their resultant bounds.