Cross-Ratio and Radical Axis: Core Invariants in Olympiad Geometry — E8 Intelligence Research

FINDING: Cross-ratio and radical axis are the two core projective/inversive invariants underlying olympiad geometry; radical axis theorem is a degenerate case of the power-of-a-point relation, and cross-ratio is the fundamental projective invariant preserved under Möbius transformations. | MATH: Cross-ratio of four collinear points (A,B;C,D) = (AC/BC)/(AD/BD) = (AC·BD)/(BC·AD). For four concyclic points, cross-ratio is real (or has argument equal to half the sum of arcs). Radical axis: locus of points P with equal powers: Pow_C1(P) = Pow_C2(P) → |P−O1|² − r1² = |P−O2|² − r2² → linear equation in P. Radical axis theorem: for three circles, the three radical axes are concurrent (radical center) or parallel. | CONNECTION: Cross-ratio invariance under Möbius transformations maps circles to circles — this is the projective backbone of the golden-ratio cyclic quadrilateral constructions. Specifically, a cyclic quadrilateral with side ratios in golden proportion (1, φ, φ², φ³) has cross-ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com