Canonical Parametric Representation of Plane Euclidean Geometry

This methodological preprint presents a systematic canonical coordinate framework for plane Euclidean configurations with two distinct reference points. A common translation and rotation preserve the geometric relations and reduce a configuration of n labelled points to 2n - 3 real parameters before additional constraints. An invariance theorem transfers proofs in canonical coordinates to the original configurations. Applications include the concurrence of the altitudes, the Euler line, the nine-point circle, and complete proofs of Simson's and Feuerbach's theorems using reusable projection and centre formulas. The manuscript organises these arguments into a common sequence of normalisation, algebraic formulation, verification, and transfer. The methodological simplification concerns the organisation of proof through a fixed coordinate frame, reusable constructions, explicit algebraic criteria, and fewer problem-specific choices of auxiliary geometric constructions. The examples illustrate this structural benefit. They do not constitute a quantitative comparison of proof length, execution time, or human cognitive effort. Normalisation by Euclidean motions and the general algebraic tools are established mathematics, and relevant prior proofs are cited. The deposit comprises two files: the English PDF manuscript and one supplementary PDF containing the complete source of an exact symbolic verifier for the framework and classical applications, its execution output, dependency information, and an English guide. An executable copy of the verifier is embedded as a PDF attachment. This is a preprint and has not undergone independent peer review. Use of generative artificial intelligence. ChatGPT assisted the English translation and final document formatting. The mathematical ideas and handwritten calculations are the authors' work. AI-assisted preparation of verification code and software-based reproducibility checks are documented separately in the supplementary material. Responsibility for the mathematical content rests with the named authors.