Incenter, Gergonne Point, and Tritangent Circles

This GeoGebra example demonstrates the incenter (I), the Gergonne point (G), and the incircle as one of the four tritangent circles tangent to all three sides of a triangle. It visually explores the relationships among angle bisectors, the incenter, contact points, and the concurrency of cevians from vertices to contact points. The construction uses GeoGebra commands: Segment to define triangle ABC, AngularBisector to find the incenter, OrthogonalLine and Intersect to locate the touch points D, E, F on the sides, Circle to draw the incircle, and lines connecting vertices to touch points to reveal the Gergonne point. This resource is ideal for high school geometry classes, helping students understand the incenter, incircle, and Gergonne point through dynamic manipulation, and can also serve as a starting point for exploring the family of tritangent circles.