Fermat Point with CAS Functions: Optimal Triple-Connection Construction

This example demonstrates how to construct the Fermat Point inside a triangle and how to compute/verify related distance relationships using GeoGebra’s CAS-driven functions. It focuses on the geometric characterization of the Fermat Point (the point whose connections to the three vertices satisfy a specific angle condition), along with the optimization intuition behind minimizing total connection length. From the construction summary, the construction is built using the GeoGebra command “Segment” to create key line segments connecting objects (e_1, e_2, e_3 and vertices A, B, C) to the FermatPoint/Fermat-related point, producing auxiliary points such as M' and intermediate points (A1, A2, B1, B2, and M). The object “Schnitt” (intersection) is used to locate crucial intersection positions needed for the construction, and a consolidated “Result” is obtained from CAS functions f, g, h to support the verification of the Fermat-point property. Suitable for high school advanced geometry, pre-university geometry labs, and learners using dynamic geometry plus CAS for proof-oriented computation.