Unifying Conic Polarity, Isogonal Conjugates, and Circular Points via Projective Involutions — E8 Intelligence Research

FINDING: Projective geometry unifies conic polarity, isogonal conjugation, and circular points at infinity via involutions on the projective line; Poncelet triangles reveal hidden conic loci for orthocenters and isogonal conjugates. MATH: - Projective plane: points at infinity form line \\( l_\\infty \\); circular points \\( I=(1:i:0), J=(1:-i:0) \\) lie on \\( l_\\infty \\). - Polarity w.r.t. conic \\( C \\): map \\( P \\mapsto p \\) (polar line) via bilinear form \\( B(P,Q)=0 \\). For a circle, this is inversion in the circle (radius \\( r \\)): \\( P' = P \\cdot (r^2/|P|^2) \\) — a projective involution. - Isogonal conjugation: in triangle \\( ABC \\), point \\( P \\) maps to \\( P^* \\) such that \\( \\angle BAP^* = \\angle PAC \\), etc. In projective terms, this is a composition of three involutions on the pencil of lines through each vertex — equivalent to a projective involution on the conic at infinity (circular points). - Poncelet triangle family: two nested conics \\( \\mathcal{E}, \\mathcal{E}_c \\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com