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.)