Geometric Construction via Substitution: Segment Lengths and Optimization Insigh

This example demonstrates how the substitution (substitution method) idea can rewrite several geometric quantities into an equivalent variable framework, making constraint-driven optimal or simplified relationships visible directly on a diagram. The key mathematical concepts include substitution to build equivalent expressions, a geometric-mean style proportional relationship, and an intuition for optimization under geometric constraints (often connected to linear/convex optimization viewpoints). In construction terms, it uses GeoGebra objects identified in the build summary—s1, l1, l2, l3, l4, n, l0, a—where l1–l4 represent the essential segment lengths (measurable geometric parts), l0 acts as a baseline/reference length for ratios or constraints, and n and a serve as variables/parameters involved in the substitution. s1 represents the substituted result or the geometric quantity computed from the re-parameterized relations. Suitable for high school through undergraduate learning, especially for teaching geometric construction combined with modeling and optimization intuition.