Geometric construction for a constant sum of squares

This GeoGebra example demonstrates an invariant phenomenon: for a specific way of partitioning segments within a triangle-based setup, the sum of the squares of certain segment lengths remains constant. The key mathematics includes Euclidean geometry constructions, triangle centers (via TriangleCenter), circle-related constraints, and expressing distance-squared relationships as algebraic identities. Construction-wise, it builds the main triangle with Polygon, generates the relevant triangle-center point(s) using TriangleCenter, creates circles and key points with Circle and Point, derives segment partitions with Segment and Midpoint, and measures lengths using Distance to form the square-sum quantities. Finally, it uses Polynomial (and function objects such as f, g, h) to translate the constant-sum condition into an algebraic form that stays consistent with the geometry. Suitable for high school through early university students, especially for lessons or explorations on invariants, distance-squared methods, and interactive geometry-to-algebra reasoning.