Centroid, Similarity Transformation and Circle in a Triangle

This GeoGebra example shows how to construct the centroid (barycenter) of a triangle and illustrate its relationship with similarity transformations and a circle. The construction begins with three arbitrary points A, B, C defining triangle ABC. Using vector subtraction, the coordinates of the centroid G are computed as the average of the vertices. A similarity transformation (dilation) with center at G and ratio 1/2 maps triangle ABC to its medial triangle formed by the midpoints, demonstrating similarity and the 1:2 ratio property. A circle centered at G is then drawn, displaying the distances from the centroid to the vertices. The key GeoGebra commands and objects used are: Point, Midpoint or Vector, Dilate (Similarity), and Circle (center and radius). This interactive file is suitable for middle‑school or high‑school geometry lessons, helping students understand the definition of the centroid, the effect of dilation as a similarity transformation, and the geometric relationship between a triangle and a circle centered at its centroid.