Geometric Construction and Locus of Segments PA + PB

This applet demonstrates the geometric construction of the sum of two segments PA + PB and explores the locus generated by varying the position of point P. Users can dynamically move point P across the plane and observe how the lengths of segments PA and PB change in real time. The applet visually traces the path (locus) formed by the sum PA + PB, helping students develop an intuitive understanding of how geometric constraints produce different curve families such as ellipses, lines, or other conic sections depending on the fixed positions of points A and B and any additional conditions imposed. The applet incorporates circles, polygons, and angle measurements alongside the core construction, enriching the exploration with multiple geometric elements. Students can investigate how the relative placement of A, B, and P influences the resulting locus, reinforcing key concepts in coordinate geometry, Euclidean constructions, and analytic geometry. By adjusting parameters interactively, learners can test hypotheses about the shape and properties of the locus, compare their predictions with the visual output, and deepen their grasp of the relationship between algebraic expressions and geometric figures. This interactive visualization is particularly valuable for understanding classical geometry problems involving minimal distance sums, reflection principles, and ellipse definitions. It bridges the gap between abstract theorem-based reasoning and concrete geometric intuition, making it an effective teaching and learning tool for secondary school and early undergraduate mathematics courses.