I'm currently reading the all-familiar "Concepts of Modern Mathematics" by Ian Stewart. At some point in his chapter on motion geometry, Stewart states:
"It is a consequence of the two-dimensionality of the plane that any rigid motion is uniquely specified by what it does to a (non-degenerate) triangle, whence it is sufficient to consider only triangles."
He follows this up with a note under the "Notes" section, saying:
This statement should perhaps be expanded. Given three points A, B, C (distinct and non-collinear) and three distances a, b, c, there is at most one point distance a from A, distance b from B and c from C. Since rigid motions do not change distances, it follows that once we know what happens to a triangle we know what happens to everything else.
Why? Doesn't a simple line segment between two points suffice to both demonstrate and specify what happens when the three basic rigid motions of translation, rotation and reflection are applied?
A line segment is not enough to distinguish the identity transformation from a reflection over the line containing the segment.
Adding a third point not collinear with the endpoints of the line segment allows you to distinguish between the two.