A new approach to euclidean plane geometry based on projective geometric algebra

The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA), designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. We explore in detail the geometric product involving pairs and triples of basic elements (lines and points), and show it encodes a wealth of fundamental metric and non-metric properties, whereby the seamless integration of euclidean and ideal aspects is particularly noteworthy. We apply the algebra to a variety of familiar topics in plane euclidean geometry, and show it compares favorably with other approaches in regard to completeness, compactness, practicality, and elegance. Numerous figures accompany the text. For readers with the requisite mathematical background, a self-contained coordinate-free introduction to the algebra is provided in an appendix.