Equations of Tangents to Circles via Geometric Construction

This GeoGebra example demonstrates how to construct and obtain the equations of tangents to a circle (typically from a given point/configuration) while visually linking the algebraic tangent equations with the underlying geometry. It centers on the tangent-to-circle property (radius is perpendicular to the tangent at the touch point), and the translation of that geometric condition into coordinate-geometry style equations. The construction is organized using GeoGebra commands Point, Intersect, and If: Point is used to create the key geometric objects (including several labeled/structured points and auxiliary points involved in the configuration), Intersect is used to compute intersection results that identify tangent-related points/lines, and If selects different construction branches based on conditions so the tangent equation/geometry remains consistent across cases. The activity is suitable for advanced high-school learners and early university students studying analytic geometry, circle geometry, and (as suggested by the example theme) complex-number or similarity-transformation viewpoints.