Bizarre Trisection (V3) = GoGeometry Action 146!

Creation of this resource was inspired by a problem posted by Antonio Gutierrez (GoGeometry). Goal here is to TRISECT the segment shown below. That is, our goal is to cut this segment into 3 congruent segments. Slide the slider slowly. As you do, carefully observe what happens. You can move the 2 LARGE POINTS (segment endpoints) anywhere you'd like at any time. How can we formally prove why this method of segment trisection actually works?

Quick (Silent) Demo