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GoGeometry Action 6!

Creation of this applet was inspired by a tweet from Antonio Gutierrez. In this applet, the smaller circle is tangent to both the larger circle and the diameter of the larger circle. Interact with this applet for a few minutes. As you do, be sure to change the locations of any 1 (or more) of the LARGER WHITE POINTS. How can you formally prove what this applet informally illustrates? (For an algebraic interpretation of the length of the purple segment, click here.)