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 .
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.