Incenters, Circumcenters, and Centroids of a Triangle

This interactive GeoGebra example demonstrates the locations of the incenter, circumcenter, and centroid of a triangle, allowing students to explore how these three special points change as the triangle's shape varies. The core mathematical concepts involve angle bisectors, perpendicular bisectors, medians, and their intersection properties, which define the incenter (intersection of angle bisectors), circumcenter (intersection of perpendicular bisectors), and centroid (intersection of medians). The construction uses GeoGebra commands such as Polygon to create the triangle, AngularBisector for angle bisectors, LineBisector for perpendicular bisectors, Midpoint and Ray to construct medians, and Intersect to find the points of concurrency. Additional commands like Angle, Segment, Circle, and OrthogonalLine are used to display angles, sides, the circumcircle, and perpendicular relationships. This resource is suitable for middle or high school geometry classes, helping students visualize the properties of triangle centers and understand geometric constructions through dynamic manipulation.