Perpendicular Bisector

Dependencies: Segment, Perpendicular, Midpoint. A derived definition: the line through the midpoint of a segment and perpendicular to it.

Definition

Let $AB$ be a segment, and let $M$ be its midpoint.

The line through midpoint $M$ and perpendicular to $AB$ is called the perpendicular bisector of segment $AB$.

Both requirements are essential: passing through the midpoint centers the line, and perpendicularity fixes its direction. Together they make the line an axis of symmetry exchanging $A$ and $B$.

Remarks

• Locus of points equidistant from the endpoints. Every point on the perpendicular bisector is the same distance from $A$ and $B$, and every point equidistant from them lies on it (Perpendicular bisector ⇔ equidistant from endpoints, Perpendicular bisector test).
• Difference from a midpoint. Equidistance alone describes the whole perpendicular-bisector line. Intersecting that line with segment $AB$ gives the single midpoint.
• The three perpendicular bisectors of a triangle are concurrent. They meet at the circumcenter, which is equidistant from all three vertices and is the center of the circumcircle (Three perpendicular bisectors meet (circumcenter)).