Hexagonal Surprise! (GoGeometry Action 85)

Creation of this applet was inspired by a [url=https://twitter.com/gogeometry/status/891435338551492608]tweet[/url] from [url=https://twitter.com/gogeometry]Antonio Gutierrez[/url]. [br][br][b]The phenomenon you're about to witness works for ANY HEXAGON (convex or concave). [br][/b]Thus, feel free to move any of the 6 white vertices below. [br][br][b][color=#a64d79]How can we formally prove what is informally illustrated here? [/color][/b]

Information: Hexagonal Surprise! (GoGeometry Action 85)