In this series of posts, we are taking a look at how to develop the congruence theorems found in the CCSSM using a transformations-based approach.

In a previous post, we showed that the mirror line for a segment \overline{AB} has a special property: each point on the mirror line is equidistant from the endpoints of the segment. In this post, we’ll examine the converse of this theorem: if we have a point O that is known to be equidistant from two points A and B, does this mean that O has to be on the mirror line for A and B?

equidistance 4.PNG

It seems intuitive that the bisector of \angle AOB will also act as a mirror for A and B.

equidistance 3.PNG

Point O is equidistant from A and B, and line OC bisects \angle AOB.

Click here to interact with this figure in GeoGebra.

The plan for this proof is to show that \overline{OB} is the mirror image of \overline{OA} in line m. To establish this, we’ll show that these two segments 1) have a common initial endpoint, 2) have the same length, and 3) have terminal endpoints that lie on a common half-line. As long as these conditions are met, we can be sure that these two segments coincide. (The technical machinery that makes this argument work is the Ruler Postulate.)

  1. Let OC be the bisector of \angle AOB, and let m be the line through O and C; let r be the reflection over line m. We’ll show that the endpoints of segment \overline{AB} are symmetric with respect to line m.
  2. Do r(\overline{OA}) and \overline{OB} have a common initial endpoint?
    1. Yes – since O is on line m, r(O) = O.
  3. Do r(A) and B lie on a common half-line?
    1. Yes – since m is the bisector of \angle AOB, the Side-Switching Theorem shows that r(\overrightarrow{OA}) = \overrightarrow{OB}, so r(A) lies somewhere on ray \overrightarrow{OB} (as does point B).
  4. Do r(\overline{OA}) and \overline{OB} have the same length?
    1. Yes – we are given that \overline{OA} = \overline{OB}, and since reflections preserve distance, r(\overline{OA}) = \overline{OB} as well.
  5. It now follows that r(\overline{OA}) = \overline{OB}. In particular, r(A) = B. This implies that line m is the mirror line for A and B. Since point O is on line m, this completes the proof that O is on the mirror line for A and B.

This proof makes use of 1) the definition of reflection, 2) the Side-Switching Theorem for angles, and 3) the fundamental assumption that reflections preserve distance.

We are now in a position to completely characterize the points on the perpendicular bisector of a segment, thus fulfilling the final requirement of Standard G-CO.9:

The points on the perpendicular bisector of a segment are exactly those that are equidistant from the segment’s endpoints.

This theorem has many applications. In particular, it will play a critical role in our proofs of the Isosceles Triangle Theorem, the Segment Congruence Theorem, and the Side-Side-Side criterion for triangle congruence.

The Equidistance Theorem is also discussed at Illustrative Mathematics, although at present the logic is problematic, as one of the commenters points out.