Rolling on a Line 3: Circle and Incircle of Right Triangle

This applet demonstrates a circle rolling along a straight line while simultaneously constructing and exploring the incircle of a right triangle. The animation visually connects the rolling motion with the fundamental geometric properties of right triangles and their incircles. As the circle rolls, the applet tracks and displays the trajectory of key points, illustrating the cycloidal curves that emerge from pure rolling without slipping. The construction incorporates rotations and translations of points, line segments, circles, and polygons, allowing users to observe how these geometric transformations interrelate. The right triangle is dynamically constructed so that its incircle interacts with the rolling process. Students can manipulate the configuration using point dragging and parameter adjustment to see how changes in the triangle's dimensions affect both the rolling trajectory and the position of the incircle. The applet highlights important relationships such as the fact that the distances from each vertex to the points of tangency on adjacent sides are equal, and that the inradius of a right triangle with legs a and b and hypotenuse c satisfies r = (a + b - c) / 2. Key learning outcomes include understanding the geometry of rolling motion, appreciating the connection between linear and rotational displacement, recognizing the invariant properties of the incircle under transformation, and visualizing how loci and trajectories are generated through geometric constraints. This interactive tool is particularly valuable for students studying Euclidean geometry, circle properties, and kinematic geometry. It supports inquiry-based learning by allowing learners to form conjectures, test them dynamically, and develop deeper intuition about the elegant relationships between triangles, circles, and motion in the plane.