Construction and Locus of a Circle Tangent to a Line and Passing Through a Fixed

This applet demonstrates the geometric construction of a circle that passes through a fixed point and is tangent to a fixed line, along with the locus traced by the center of such a circle. The construction begins by establishing a fixed point (the focus) and a fixed line (the directrix). Using these elements, the applet constructs angle bisectors and their intersections to determine the possible positions of circle centers that satisfy both conditions simultaneously. As the user drags the defining points, the circle dynamically changes size and position while maintaining tangency to the line and passing through the fixed point. The locus of the circle's center forms a parabola, which is also displayed in the applet. This provides a vivid, visual confirmation of the fundamental geometric definition of a parabola: the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). Each constructed circle center lies on this parabola because it is equidistant from the fixed point and the tangent line by construction. Users can interact with the applet by dragging the fixed point, the tangent point on the line, or other key elements to observe how the circle and its center adapt in real time. This interactivity allows students to explore a wide range of configurations and verify that regardless of the chosen tangent point, the center always remains on the same parabola. This applet is particularly valuable for students learning analytic geometry and conic sections. It bridges the gap between the abstract algebraic definition of a parabola and its concrete geometric realization. By watching the dynamic construction unfold, students develop an intuitive understanding of why the parabola is defined as an equidistance locus and how tangent-circle constructions naturally lead to this curve. The applet serves as an excellent supplement to lessons on locus problems, the geometric properties of conics, and the connections between synthetic and coordinate geometry.