Equivalent Polar Coordinate Representations of Points

This GeoGebra example demonstrates how the same point in the plane can be represented by equivalent polar coordinates using different radius/angle choices (notably angle periodicity differing by multiples of 2π), and it verifies the equivalence visually through construction. Key concepts include polar coordinate representation, the equivalence of angles via periodicity, and the geometric relationship between radius/angle and the corresponding point location. The construction leverages GeoGebra commands such as Ray (for angular direction), Circle (for radius constraint), Mirror (to reflect/relate positions), RandomElement (for controlled random selection), FractionText and Textfield (for readable symbolic/interactive labeling), and If (to switch between cases). The setup uses multiple geometric objects and parameter lists/variables (e.g., points like A, B, C, D and mirrored variants, along with f_{1}, g_{1}, and lists such as flist/rlist/ra), plus InputBox1–6 for user input and buttons (button1, button2) to trigger different verification steps. Suitable for high school to early undergraduate learners in coordinate geometry and trigonometry, especially when teaching polar coordinates and parameter-based graphing/validation.