Triangle and Equilateral Triangles

① This example demonstrates constructing equilateral triangles on the sides of an arbitrary triangle and exploring the resulting complex geometric configuration. ② Core mathematical concepts include equilateral triangles, congruence, midpoints, ellipses, expected value, and median, blending geometry with statistics. ③ Construction details: using GeoGebra commands such as Polygon to create the initial triangle poly1, Midpoint for midpoints, Segment for connecting segments, and Intersect for finding intersections; multiple equilateral triangles (poly2, poly3, poly4) are built, with auxiliary points (D, E, F, etc.) and segments (f, g, h, etc.) to complete the figure; also includes ellipses and bar charts for statistical representation. ④ Suitable for middle and high school geometry classes, math labs, or inquiry-based learning, helping students understand properties of equilateral triangles, congruence, and the integration of geometry with statistics.