A graphic method for constructing the midpoint trajectory between a point on a circle and any point on the geometric sketchpad

Graphic method for constructing the midpoint trajectory between a point on a circle and any point on the geometric sketchpad In mathematics teaching, we often encounter the problem of constructing the midpoint trajectory of a point on a circle and any point on the geometric sketchpad. This question seems simple, but in fact it contains a profound connection between geometry and algebra. It is a classic question in geometry. This article will explore the solution to this problem from both theoretical and practical aspects, hoping to provide some ideas and methods for teachers and students. In the following content, we will first introduce the background and related knowledge of this problem from a theoretical level, and then combine it with specific examples to demonstrate how to use the geometric sketchpad to construct such a trajectory.

Construct a circle O and pick a point A on the circle. Open the geometric sketchpad, select the [Circle Tool] on the left sidebar and draw a circle O in the working area. Use the [Point Tool] to pick any point A on the circle.

Pick any point B and construct line segment AB. Use the [Point Tool] to draw a point B arbitrarily in the blank area, then select points A and B in sequence, click the [Construct] menu on the upper menu bar, and select the [Line Segment] command from its drop-down option, thus constructing the line segment AB.

Construct the midpoint M of line segment AB. Use the move tool to select line segment AB, click the [Construction] menu on the upper menu bar, and select the [Midpoint] command from its drop-down option, thus obtaining the midpoint M of the line segment.

Construct the trajectory. Use the move tool to select point A and point M in sequence, click the [Construction] menu in the upper menu bar, and select the [Trajectory] command from the drop-down option. This way you will get the motion trajectory of midpoint M,

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