Towards a Systematic Encyclopedia of Special Triangles — Fifth Edition

This work presents the foundational framework for a comprehensive taxonomic encyclopedia of special triangle families in Euclidean geometry. Each family is defined by a one-dimensional locus, expressed as an explicit algebraic equation in a unified Cartesian coordinate system with fixed vertices A = (1, 0) and B = (−1, 0). The classification system is governed by seven strict criteria: geometric or algebraic-geometric nature, invariance under similarity transformations, purely theoretical character, restriction to the planar Euclidean setting, one-dimensionality of the locus, formulation of properties as logical equivalences, and the requirement that each classification generates genuine new geometric structure. The catalogue covers six chapters of triangle families: angular families (right, semi-right, 60°, 120°, 30°, 45°, angle-multiple, and trigonometric families), side-based families (linear, quadratic, and harmonic relations), line-related families (parallelism, orthogonality, collinearity, and concurrency), circle-related families (tangency, concyclicity, and orthogonality of circles), triangle-center relations (distances and coincidences of classical centers), and miscellaneous families. Each entry includes a defining condition, a complete chain of equivalent properties involving triangle centers, notable lines, and associated circles, and an explicit locus equation. Triangle center notation follows Clark Kimberling's Encyclopedia of Triangle Centers (ETC). Keywords: Euclidean geometry, triangle classification, geometric loci, triangle centers, algebraic curves, special triangles, taxonomy, Encyclopedia of Triangle Centers