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 \\(\\mathcal{E}\\) (outer) and \\(\\mathcal{E}_c\\) (inner caustic) be nested ellipses. For a Poncelet triangle family inscribed in \\(\\mathcal{E}\\) and circumscribed about \\(\\mathcal{E}_c\\): (i) Orthocenter locus: \\(\\mathcal{H} = R_{90^\\circ}( \\mathcal{E} )\\) scaled by factor \\(k\\) (homothety), i.e., \\(\\mathcal{H} \\sim \\mathcal{E}\\) rotated by \\(\\pi/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 \\((\\mathcal{E}, \\mathcal{E}_c)\\). | CONNECTION: The \\(90^\\circ\\) 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