Isogonal Conjugate Loci and Orthocenter Homothety in Poncelet Triangles — E8 Intelligence Research

FINDING: Locus of isogonal conjugate of a fixed point over Poncelet triangles (nested ellipses) is a conic, and orthocenter locus is a rotated, homothetic copy of the caustic ellipse. | MATH: Let \(E\) (outer) and \(E_c\) (inner caustic) be nested ellipses. For a Poncelet triangle family inscribed in \(E\) and circumscribed about \(E_c\): (i) Orthocenter locus: \(H = R90^∘( E )\) scaled by factor \(k\) (homothety), i.e., \(H ~ E\) rotated by \(π/2\). (ii) Isogonal conjugate locus of fixed point \(P\): a conic (not generally an ellipse; may be hyperbola/parabola depending on \(P\) and caustic). No explicit equation given in abstract, but the conic is determined by \(P\) and the pair \((E, E_c)\). | CONNECTION: The \(90^∘\) rotation links to the imaginary circular points at infinity \(I=(1:i:0), J=(1:-i:0)\) — the orthocenter is the isogonal conjugate of the circumcenter, a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com