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 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 that is known to be equidistant from two points and , does this mean that has to be on the mirror line for and ?

It seems intuitive that the bisector of will also act as a mirror for and .

Point is equidistant from and , and line bisects .

Click here to interact with this figure in GeoGebra.

The plan for this proof is to show that is the mirror image of in line . 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.)

- Let be the bisector of , and let be the line through and ; let be the reflection over line . We’ll show that the endpoints of segment are symmetric with respect to line .
- Do and have a common initial endpoint?
- Yes – since is on line , .

- Do and lie on a common half-line?
- Yes – since is the bisector of , the Side-Switching Theorem shows that , so lies somewhere on ray (as does point ).

- Do and have the same length?
- Yes – we are given that , and since reflections preserve distance, as well.

- It now follows that . In particular, . This implies that line is the mirror line for and . Since point is on line , this completes the proof that is on the mirror line for and .

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.

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