Bryan Johnson's Triangle Theorem

Creation of this resource was inspired by a comment from Bryan Johnson in response to this tweet originally inspired by Antonio Gutierrez (GoGeometry). See pic below. Link to the original GeoGebra resource (special case of this illustration where 3 equilateral triangles are built off the sides of this shaded triangle can be found here).
In the GeoGebra applet below, the triangles built off the sides of the shaded triangle are all similar to each other. You can control the size of one angle of these similar triangles by adjusting the slider in the middle. The LARGE POINTS are MOVEABLE. The slider off to the left controls the size of the lower left angle within the shaded triangle. How can we formally prove what is dynamically illustrated here?

Quick (Silent) Demo