Midpoint

Dependencies: Ruler axiom, Distance. A derived definition: combine “between,” supplied by the ruler axiom, with equal distances to identify the center of a segment.

Definition

Let $A$ and $B$ be distinct points, and let $M$ lie on segment $AB$.

If $M$ lies between $A$ and $B$ and $|AM|=|MB|$, then $M$ is called the midpoint of segment $AB$.

“Between” keeps $M$ on the segment rather than an extension, while equality of the two distances places it exactly in the middle. Both conditions matter.

Remarks

• Existence and uniqueness. Under ruler coordinates, the point with coordinate $\tfrac{x_A+x_B}{2}$ lies between the endpoints, is equidistant from them, and is unique.
• Equidistance alone is not enough. The set of all points satisfying $|MA|=|MB|$ is the perpendicular bisector of $AB$ (Perpendicular bisector ⇔ equidistant from endpoints). Its intersection with segment $AB$ is the midpoint.
• Midpoints begin many constructions. They lead to triangle medians, midsegments (Triangle midsegment theorem), and point symmetry. Many elementary geometry constructions start by taking a midpoint.