The Zariski Closure of a 3-Ellipse

①This example demonstrates the Zariski closure of a 3-ellipse, defined as the set of points whose sum of distances to three fixed foci (points A, B, C) is constant, revealing the complete algebraic curve in the plane. ②Core mathematical concepts include generalized ellipses (multi-focal ellipses), algebraic curves, Zariski closure (the smallest algebraic variety containing a given set), and locus equations. ③Construction highlights: using GeoGebra commands Segment and LocusEquation, with objects A, B, C, P (moving point), segments a, b, c, auxiliary points Q, R, and parameter s, to generate the locus equation eq1. ④Suitable for high school to undergraduate geometry and algebra courses, particularly for analytic geometry, introductory algebraic geometry, or mathematical experimentation, helping students understand the relationship between locus equations and algebraic closures.