Number of Tangents to a Circle from a Point

This GeoGebra example demonstrates how the number of tangent lines that can be drawn from a point to a circle depends on the point's location relative to the circle: two tangents from an exterior point, one tangent from a point on the circle, and no tangents from an interior point. Core mathematical concepts include the definition of a tangent to a circle, perpendicularity between tangent and radius at the point of tangency, comparison of point-to-center distance and radius, and the positional relationship between a line and a circle. The construction uses GeoGebra commands Circle, Tangent, and Intersect with objects such as A_1, B, circle, A_2, a_1, b, d, D, E, A, a, and c, allowing dynamic observation as point A moves. Suitable for middle school geometry instruction on circles and high school analytic geometry review or classroom demonstration.