Symmetry via Transformations: Circles/Spheres with Scatter Construction

This example demonstrates how to construct symmetry-related structures involving circles and spheres in a geometric setting by using systematic transformations (rotation, dilation, translation) together with intersection and line/path tools. It highlights the emergence of symmetric configurations and how geometric relations can be measured and visualized through scatter-like point sets and angle information. Key mathematical concepts include properties of circles and spheres, symmetry created by geometric transforms, vectors and lines for directional relationships, and angle measurement for quantitative constraints, with continuity ensured by consistent geometric construction. Construction approach: IntersectPath and Plane are used to determine intersection-based positions; Rotate, Dilate, and Translate apply the core transformations in 2D/3D; Segment, Line, and Polygon organize the underlying geometric entities; Angle and Vector extract measurable angle and direction; Point plus Textfield/labeling track key parameters and corresponding points (including paired/derived point sets) that reveal the symmetry mapping. Suitable for upper-secondary students and early undergraduate learners in transformation geometry, GeoGebra-based construction practice, and exploratory math modeling.