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 \( l_∞ \); circular points \( I=(1:i:0), J=(1:-i:0) \) lie on \( l_∞ \). - Polarity w.r.t. conic \( C \): map \( P ↦ p \) (polar line) via bilinear form \( B(P,Q)=0 \). For a circle, this is inversion in the circle (radius \( r \)): \( P' = P · (r^2/|P|^2) \) — a projective involution. - Isogonal conjugation: in triangle \( ABC \), point \( P \) maps to \( P^* \) such that \( ∠ BAP^* = ∠ PAC \), 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 \( E, E_c \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com