Geometric Mean Cameo Appearance within a Parabola

Creation of this resource was inspired by a tweet posted by Bryan Penfound. Feel free to move the LARGE BLUE POINT and LARGE ORANGE POINT anywhere you'd like. Slide the slider slowly. As you do, pay close attention to what you observe here. How would you describe the phenomena you see in your own words? How can we formally prove what is dynamically illustrated here?