Comparing Graphs Through Geometric Constructions and Barycenters

This example demonstrates how to compare geometric graphs/configurations by establishing correspondences between key parts of the figure and then validating the relationships using symmetry and barycenter (centroid) ideas. It focuses on core concepts such as 2D vectors in a geometric setting, triangle-based constructions (including equilateral/related triangle structures), kite-like quadrilateral geometry, and the location/role of the centroid or barycenter in confirming the comparison. The construction is driven by a set of labeled points—A2, B1, C1, D1, E1, F1, G1, H1, and A1—whose positions are created and then used to form the relevant triangles/polygons and to compute or reason about the barycenter and its geometric/vector relations. Suitable for middle to high school geometry enrichment, or for first-year undergraduate students learning geometric construction, coordinate geometry, and barycenter-based verification.