Investigating the Ellipse: Foci, Distances, and Geometric Properties

① This interactive GeoGebra example dynamically demonstrates the fundamental property of an ellipse: the sum of distances from any point on the ellipse to its two foci is constant, with live measurements of segment lengths. ② Core mathematical concepts include the definition of an ellipse (foci, major and minor axes), distance formulas, geometric measurement, and auxiliary constructions involving equilateral triangles and reflections. ③ Construction details: using GeoGebra commands such as Segment to draw lines, LaTeX for typesetting formulas, and Distance to compute lengths; objects include center, Focus1, Focus2, moving point M, auxiliary points P, Q, R, S, major axis Major, minor axis Minor, and text labels (distancePQ, TextFocus1M, etc.) that display dynamic values. ④ Suitable for high school or early college courses in analytic geometry and calculus, ideal for classroom demonstrations or student-led exploration of ellipse definitions and properties, as well as a teaching example for geometric construction and dynamic measurement.