Locus of a Moving Circle Tangent to Two Fixed Circles

Locus of a Moving Circle Tangent to Two Fixed Circles This GeoGebra applet demonstrates the geometric properties and locus of a moving circle that maintains both external and internal tangency with two fixed circles simultaneously. Through interactive constructions involving points, circles, intersection points, ellipses, line segments, and rays, users can visually observe and explore how the center of the moving circle traces out conic sections depending on the tangency configuration. When the moving circle is externally tangent to both fixed circles, the locus of its center forms an ellipse. This arises because the sum of the distances from the moving center to each fixed center remains constant, which is precisely the defining geometric property of an ellipse. When the moving circle is externally tangent to one fixed circle and internally tangent to the other, the locus becomes a hyperbola. In this case, the absolute difference of the distances from the moving center to the two fixed centers is constant, matching the geometric definition of a hyperbola. The applet allows users to drag the moving circle and watch in real time as its center traces out the corresponding curve. This dynamic interaction helps students connect the algebraic and geometric definitions of conic sections to a concrete construction problem. By manipulating the positions and radii of the two fixed circles, learners can observe how the shape and size of the resulting conic change, reinforcing the relationship between the parameters of the fixed configuration and the properties of the locus. This interactive exploration supports the study of analytic geometry at the high school level by bridging classical geometric construction with modern coordinate methods. It provides a visual proof of the conic definitions based on distance relationships, offering an alternative perspective to the standard focal definitions found in textbooks. The applet is particularly suited for classroom demonstrations, guided inquiry activities, and independent student exploration of circle tangency conditions and their connection to ellipses and hyperbolas. Students benefit from this applet by developing a deeper intuitive understanding of why conic sections arise naturally from simple tangency constraints, strengthening their ability to reason geometrically and to make connections between seemingly unrelated mathematical concepts.