# Inscribing a segment in a triangle with a given midpoint

- Author:
- 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?