Golden Ratio Invariants in Poncelet 3-Periodics with Circular Caustics — E8 Intelligence Research

FINDING: Poncelet 3-periodics inscribed in an ellipse and circumscribing a circular caustic exhibit invariant center-power, fixed barycenter/Nagel-line loci, and an "orthofocal" circumellipse whose center and focus structure link to the golden ratio. | MATH: For concentric ellipse E(a,b) and circle K(C,r), center power w.r.t. circumcircle and Euler circle is invariant: \( Pow(C, circumcircle) = const \), \( Pow(C, nine-point circle) = const \). The orthofocal circumellipse has one focus at X(4) (orthocenter); its center is numerically found to be a fixed point (likely the homothety center of E and K). For the circular-caustic family, the barycenter locus is a fixed point (constant), and the Nagel line sweeps a fixed direction. | CONNECTION: The invariant center power implies a constant ratio between radii of circumcircle and incircle (caustic) — for the special case where the outer ellipse degenerates to a circle, the Poncelet triangles become Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com