Isogonal Conjugation as Projective Involution on Circumconic Line Pencils — E8 Intelligence Research

FINDING: Isogonal conjugation induces an involution on the pencil of lines through a point, with fixed lines determined by a circumconic's intersection with the pencil — a projective structure linking triangle geometry to conic involutions. | MATH: Isogonal conjugation in a triangle is a projective involution on the pencil of lines at a vertex (order 2, no fixed lines generically). For a circumconic, the map sending a line to its isogonal conjugate's intersection with the conic yields a projective involution on the conic's points. Fixed lines of this involution correspond to tangents from the point to the conic (or degenerate cases). Key invariant: cross-ratio of four lines in the pencil is preserved under isogonal conjugation (since it's a projective involution). The arxiv paper (1912.08330) shows: for two pairs of isogonal conjugates, the spiral center lies on a fixed circle; when two circumconics are tangent at two points, the induced involution on the pencil has fixed lines at thos Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com