Square and Quarter Circle Action = GoGeometry Action 81!

Creation of this applet was inspired by a [url=https://twitter.com/gogeometry/status/890591152986042368]tweet[/url] from[url=https://twitter.com/gogeometry] Antonio Gutierrez[/url] (GoGeometry.) [br][br][color=#9900ff][b]How can we formally prove what this applet informally illustrates?[/b][/color] [br][br]

Information: Square and Quarter Circle Action = GoGeometry Action 81!