Locus of the Midpoint of a Segment from an External Point to a Circle

This applet demonstrates how to find the locus of the midpoint of a segment connecting a fixed point outside a circle to a variable point moving along the circle. Let A be a fixed point located outside a given circle, and let P be a point that moves freely along the circumference of the circle. The midpoint Q of the segment AP is constructed, and as P traverses the entire circle, the path traced by Q forms a new circle. This new circle has exactly half the radius of the original circle and is centered at the midpoint of the segment connecting the original circle's center to the external point A. The applet makes use of GeoGebra's locus command to dynamically and visually reveal this result in real time, allowing students to observe how the midpoint transforms the original circle into a smaller, concentrically related circle. The construction integrates several key plane geometry concepts, including circles, line segments, midpoints, and loci, providing an interactive and inquiry-based learning experience. By dragging point P or adjusting the position of the external point A, students can explore how the locus changes in response to different configurations, deepening their understanding of geometric transformations and locus problems. This applet is well suited for classroom instruction or independent exploration in topics covering coordinate geometry, transformational geometry, and the properties of circles. It serves as an excellent visual aid for helping students develop intuition about why the locus of midpoints between a fixed point and a moving point on a circle produces another circle, and it reinforces the connection between algebraic reasoning and geometric visualization.