Move one vertex around and the centroid draws a small circle
The three vertices of a triangle lie on a large circle. Keep two points (B and C) fixed, and move the remaining point (A) once around the circle. Predict what shape the path traced by the triangle’s centroid (the average of the three vertex coordinates) will make, then check it yourself.
Why does it become a circle? The centroid is G = (A+B+C)/3. Since B and C are fixed, (B+C)/3 is also a fixed point. When A moves on a circle of radius R, A/3 moves on a circle of radius R/3. Therefore, G also traces a circle of radius R/3 — a circle exactly one-third the size of the large circle.