Locus of an Ellipse: Geometric Construction with Tangent Circles

This interactive GeoGebra applet provides a vivid geometric demonstration of how an ellipse is formed as a locus, using the elegant construction of tangent circles. The applet features two fixed circles of different sizes and a dynamically moving circle that remains simultaneously tangent to both. As the moving circle adjusts its position and radius to maintain tangency with the larger and smaller fixed circles, its center traces out a perfect ellipse, offering students a concrete visual pathway to understanding one of the fundamental definitions of conic sections. The core mathematical principle at work is the focal property of an ellipse. The centers of the two fixed circles serve as the foci of the resulting ellipse. Because the moving circle is tangent to both fixed circles, the sum of the distances from the center of the moving circle to the two fixed centers remains constant throughout its motion. This constant sum is precisely the defining characteristic of an ellipse, and the applet makes this abstract algebraic condition tangible through direct geometric observation. The construction incorporates several foundational geometric objects, including circles, points, line segments, angle bisectors, and intersection points, all working together to maintain the tangency constraints as the configuration changes. Students can interact with the applet by dragging key points to reposition the moving circle along its path, watching in real time as the locus is generated point by point. They can also modify the sizes and positions of the fixed circles to explore how these parameters affect the shape, orientation, and eccentricity of the resulting ellipse. This applet serves as an excellent pedagogical tool for secondary and introductory college-level courses in analytic geometry and precalculus. It bridges the gap between the algebraic equation of an ellipse and its geometric meaning, helping students develop deeper spatial intuition. By observing the construction dynamically, learners can discover on their own why the tangent circle method produces an ellipse, reinforcing their understanding of loci, tangency conditions, and the focal definition of conic sections. The visual and interactive nature of the applet makes it particularly effective for inquiry-based learning, encouraging students to experiment, conjecture, and verify geometric relationships independently.