Comparing Arithmetic and Geometric Mean

This GeoGebra example dynamically compares the arithmetic mean (AM) and geometric mean (GM) of two positive numbers, visually demonstrating the classic inequality AM ≥ GM. The core mathematical concepts include arithmetic mean, geometric mean, semicircle, perpendicular diameter theorem, and the use of logarithmic functions for comparison. The construction uses GeoGebra commands such as Semicircle, Segment, Point, OrthogonalLine, Intersect, and Midpoint. Key objects include points A and B representing the two numbers, semicircle t, diameter f, foot D, auxiliary lines g, h, i, intersection H, segment lengths a and b, perpendicular j, lengths ma and mg for the means, text labels text1 through text4, midpoint M, and auxiliary object k. This is suitable for high school and above students learning inequality proofs, geometric intuition, and function graphing, as well as for classroom demonstrations by teachers.