Equilateral Triangle Emergence in a Rigid Area-Normalized Overlay of Minas Gerais and Egypt

This preprint investigates a rigid area-normalized geometric overlay between the territorial polygon of Minas Gerais (Brazil) and the territorial polygon of Egypt under strictly constrained transformations (uniform scaling, rigid rotation, and translation). Minas Gerais is historically represented by an equilateral triangle in its state flag. In this study, however, no triangular symmetry is imposed a priori. The optimization procedure first searches freely for the largest inscribed triangle within the main connected component (“Core”) of the polygon intersection. At the global optimum (θ* ≈ 144°), the returned triangle is equilateral (60°/60°/60°). A fully audited replication using simplified polygons yields Lm ≈ 705 km, with published side lengths, angular deviations, and geodesic verification on the sphere (Appendix A.3). A higher-resolution run reports consistent orientation and scale (Lm ≈ 617 km), summarized separately (Appendix A.4). The manuscript establishes a strictly geometric result and does not claim historical intent, symbolic causality, or astronomical interpretation.