Building Similar Cones: Testing Similarity with Dilation

This GeoGebra activity demonstrates how to construct a pair of right circular cones and then test whether the cones are similar by interactively modifying their defining geometry. It focuses on the core idea of similarity in solid geometry and uses a dilation (similarity transformation) to create a candidate cone that should correspond under proportional scaling. Key concepts include similarity, correspondences between geometric elements, and using a scale factor to preserve shape relationships. Construction-wise, the model uses PointIn to place defining points, then Point, Line, and Segment to build the cone’s generatrix/axis structure, Surface to render the cone solids, and Intersect to compute critical intersection points that enforce the configuration. The command Dilate is used to apply proportional scaling to relevant elements, and Max (together with auxiliary quantities/flags such as check) supports an automated verification of the similarity conditions. Suitable for secondary advanced geometry, exploratory classroom work, and introductory 3D geometry instruction.