Proof Without Words of the AM-GM Inequality

This GeoGebra applet provides an elegant visual demonstration of the Arithmetic Mean-Geometric Mean (AM-GM) inequality through a classic proof without words. The AM-GM inequality states that for any two non-negative real numbers a and b, their arithmetic mean is always greater than or equal to their geometric mean, expressed algebraically as (a + b) / 2 is greater than or equal to the square root of ab. Rather than relying on algebraic manipulation, this applet reveals the truth of this fundamental inequality through pure geometric reasoning. The construction is built around a semicircle whose diameter represents the sum of two positive quantities, a and b. A point on the diameter divides it into two segments of lengths a and b. From this dividing point, a perpendicular line is drawn upward to intersect the semicircle. The length of this perpendicular segment corresponds exactly to the geometric mean, while the radius of the semicircle represents the arithmetic mean. By simply observing the figure, one can immediately see that the perpendicular segment is always shorter than or equal to the radius, with equality occurring only when a equals b. Users can interact with the applet by dragging the dividing point along the diameter to adjust the relative sizes of a and b. As the point moves, the semicircle and all associated segments update dynamically, allowing learners to explore countless configurations in real time. The visual comparison of segment lengths makes the inequality intuitively obvious without requiring any symbolic computation or formal algebraic proof. This applet is particularly valuable for students encountering the AM-GM inequality for the first time, as it bridges the gap between abstract algebraic statements and concrete geometric intuition. It helps learners appreciate that inequalities are not merely symbolic relationships but can be understood as spatial and visual truths. Teachers can use this tool to spark discussions about when equality holds, to introduce optimization problems that rely on the AM-GM inequality, and to encourage students to seek geometric interpretations of algebraic results. The proof without words approach also cultivates mathematical creativity by showing that rigorous arguments can be communicated through carefully constructed diagrams rather than equations alone.