Morley's Theorem Proof Sketch from Class

In this picture we are given triangle UVW. Equilateral triangle XYZ has isosceles triangles built on its edges (adding points R, S, and T) with base angles as described in the proof in the text; and points A, B, C are then located as described in text (pp. 83-85). Triangle ABC is similar to UVW, with angle trisectors meeting at the vertices of equilateral triangle XYZ. The vertices of can be dragged to see how the construction changes.

How does the proof use similarity of triangles to get the result?