Geometric Distance-Constrained Optimization Construction

This example visualizes an optimization process under geometric constraints by constructing lines and circles, measuring key point-to-reference distances, and using height/distance quantities to compare candidates and identify an optimal configuration. It centers on core ideas such as distance-based optimization, circle–line (or circle–other) intersection to determine feasible points, and a function/height relationship interpreted through geometry. Construction workflow in GeoGebra: Line for constraint lines; OrthogonalLine to build perpendiculars and obtain heights/projections; Circle and Intersect to generate intersection points that serve as candidate solutions; Segment and Polygon to assemble and display the geometric figure; Distance to compute the objective metric (e.g., distanceAF); PointIn to determine region membership; and Text labeling (e.g., TextAF, Textq1) to present key computed parameters. Suitable for high-school through introductory university use, especially for teaching optimization thinking with geometric modeling and for exploratory graph/geometry-based problem solving.