Nine-Point Circle: Midpoints, Feet of Altitudes, and Euler Points on a Common Ci

This GeoGebra example demonstrates the construction and properties of the famous nine-point circle (Euler circle or Feuerbach circle) for any triangle: the three side midpoints, the three feet of the altitudes, and the three midpoints of the segments from each vertex to the orthocenter all lie on a single circle. The core mathematical concepts include the orthocenter, centroid, circumcenter, and the relationship between the nine-point center and these triangle centers, as well as the geometric loci of midpoints and perpendicular feet. The construction uses GeoGebra commands to create nine key points (A1 through A9), with auxiliary points B1-C1, B2-C2, etc., to construct midpoints and altitude feet, then connects them via a list (list1) and a circle (c), with a text label (text1) for annotation. This example is suitable for high school and above students learning triangle centers, circle properties, and dynamic geometry exploration, and is also useful for teachers demonstrating the nine-point circle theorem visually.