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). After introducing the algebra, it presents the first detailed study of the geometric product of basic elements: pairs of lines, pairs of points, a point-line pair, 3 lines, and 3 points, with particular attention to the seamless integration of euclidean and ideal aspects. This yields a compact, powerful geometric toolkit which the article then applies to a variety of topics in plane euclidean geometry: distance formulae, sums and differences of points and of lines, isometries via sandwiches, the join operator, orthogonal projection, and a step-by-step solution of a sample geometric construction. In conclusion, the article compares the PGA approach to the analytic geometric approach and also alternative geometric algebra approaches to plane geometry. 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.