Triangle Inequality Open-Middle Construction (V2)

This example visualizes the triangle inequality and the “open middle” feasibility region by using geometric construction and labeled intersection points. It highlights the idea that for side lengths to form a triangle, the sum of two sides must exceed the third, and it shows how the valid set appears as an open middle region under length constraints. The core concepts are triangle inequality (Triangle Inequality), geometric relationships defined by circles, and the way inequality constraints translate into an admissible region through intersections. Construction uses GeoGebra commands such as Circle (circle boundaries for length constraints), Point (key control points), Segment and Polygon (to form triangle/polygon outlines), Intersect (to compute constraint-driven intersection points), and Textfield (to label critical lengths/conditions). The configuration is built from objects including control points (e.g., A, B, C, P) and auxiliary variables (e.g., a, b, c, d, h, e, k) that represent side lengths and parameters. Suitable for middle school to early university geometry and inequality instruction, as well as interactive exploration and visualization tasks.