Locus of a Driven Point: Melon-Bean Principle Demonstration

This applet demonstrates the Melon-Bean Principle, a classical concept in plane geometry that describes the relationship between the loci of a driving point and its corresponding driven point. The principle states that when a driving point moves along a given path, a driven point constructed through a fixed geometric transformation—such as rotation, scaling, or similarity—will trace out a corresponding locus, and the shape of the driven locus preserves the structural nature of the driving path. In this interactive model, students can observe how a driving point moving along a semicircular arc generates a predictable trajectory for a driven point. The applet constructs the driving point on a semicircle and defines the driven point through a fixed angular rotation and proportional distance ratio relative to a center of similitude. As the user drags the driving point along its semicircular path, the driven point moves in concert, dynamically revealing its own locus in real time. A construction using semicircles, line segments, and points makes the geometric relationship transparent and visually clear. The applet highlights several key mathematical ideas. First, it illustrates how similarity transformations—combinations of rotation and uniform scaling—map one geometric figure onto another while preserving shape. Second, it shows that the locus of the driven point is itself a circular arc whose radius and center can be derived from the transformation parameters, reinforcing the connection between algebraic ratios and geometric intuition. Third, it connects directly to common middle-school geometry problems involving locus construction, angle chasing, and optimization. Students benefit by developing spatial reasoning and an intuitive grasp of how constrained motion generates curves. By manipulating the driving point and observing the resulting traced path, learners can test conjectures, verify theoretical predictions, and build confidence in solving locus problems that appear frequently in competitions and standardized exams. The dynamic visualization transforms an abstract theorem into an exploratory experience, making the Melon-Bean Principle accessible and memorable for secondary geometry students.