Regular Pentagon Wonder: GoGeometry Action 192!

Creation of this resource was inspired by [url=https://gogeometry.com/problem/p261_regular_pentagon_inscribed_circle_distance.htm]this problem[/url] from Antonio Gutierrez. [br][b][br]How can we formally prove what is dynamically illustrated here? [/b]
The pentagon shown below is regular. The LARGE POINTS are MOVEABLE. Be sure to slide the slider SLOWLY!
Quick (silent) demo

Information: Regular Pentagon Wonder: GoGeometry Action 192!