Geometric Construction for Constrained Optimization with Lines and Circles

This example demonstrates a geometric route to constrained optimization by constructing lines and circles, locating their intersection points, and using orthogonal (perpendicular) projections to reveal how a target quantity attains an extremal configuration under given restrictions. The key math ideas include optimization (finding a minimum/maximum within constraints), analytic geometry with line–circle relationships (a geometric view related to conic-section behavior), inequality/concept of feasible regions, and distance/projection-based reasoning. It is built in GeoGebra using Line to draw the main lines, OrthogonalLine to form perpendiculars or orthogonal projections, Circle to define the circle constraint, Intersect to compute line–circle intersection points, and Segment/Polygon to assemble the significant lengths and the visual region. The construction involves many labeled points and intermediate entities (e.g., A, B, C, D, E, plus f, g and annotation objects) to guide the optimization narrative. Suitable for advanced high-school or introductory college students studying optimization ideas alongside analytic geometry and inequality-constrained reasoning.