More Geometric Mean Action! (GoGeometry Action 86)

Creation of this applet was inspired by a tweet from Antonio Gutierrez (GoGeometry). The 2 circles you see are tangent to each other. Feel free to move any of the LARGE PINK POINTS and/or LARGE GRAY POINT anywhere you'd like. How can we formally prove what is dynamically illustrated here?

Quick Demo: 0:45 - 1:24 (BGM: Andy Hunter)