Cut-The-Knot Action 19!

Creation of this applet was inspired by a tweet from Alexander Bogomolny. You can move the VERTICES of the triangle anywhere you'd like at any time. You can also change the size of the pink angle by using the pink slider. You can move the LARGE TURQUOISE POINT (soon-to-appear) as well. How can we formally prove what is dynamically illustrated here?

Quick (Silent) Demo