Inscribing a segment in a triangle with a given midpoint

Roy Wright
An intriguing question from James Tanton: Suppose X is a point inside a triangle △RST. Is X the midpoint of some line segment AB with A and B on the triangle? If so, how can we find A and B? Below, the points R, S, T, and X can be moved, and the line segment AB is shown. (There is often more than one choice for this segment.)
To construct the segment, we can find a triangle △UVW that is similar to △RST, with similarity ratio 1/2, such that X lies on △UVW and one of the sides of △UVW is contained in the corresponding side of △RST. (This is a lot easier than it sounds.) The desired point A can be found by reflecting one of the vertices of △RST across the corresponding vertex of △UVW. The desired point B is found by reflecting A across X. Why does this construction work?