Fundamental Locus Theorems

Definition

A Locus (plural: loci) is the set of all points, and only those points, that satisfy one or more specific geometric conditions. A locus is essentially the “path” or “collection” traced by a point moving according to a rule.

Why It Matters

Locus theorems allow us to define shapes not by their coordinates, but by their rules of movement; this is the foundation for everything from orbital mechanics to robotics, where we must predict and constrain the precise paths that objects take through space.

Core Concepts

• Equidistance from a Point: In a plane, the locus of points at a fixed distance $r$ from a fixed point $P$ is a circle with center $P$ and radius $r$.
  • How to read: “The distance r from point P.”
  • Meaning: The set $\{X \mid |XP| = r\}$ traces a circle — every point on it is exactly $r$ units from $P$.
• Equidistance from Two Points: In a plane, the locus of points equidistant from two fixed points $A$ and $B$ is the perpendicular bisector of the segment $\overline{AB}$.
  • How to read: “The distance from A equals the distance from B.”
  • Meaning: The set $\{X \mid |XA| = |XB|\}$ is the perpendicular bisector of $\overline{AB}$.
• Equidistance from Two Parallel Lines: The locus is a third line parallel to both and midway between them.
• Equidistance from the Sides of an Angle: The locus of points in the interior of an angle equidistant from its sides is the angle bisector.
• Locus in Space: If the condition is not restricted to a plane, the results shift dimensions (e.g., points equidistant from a point $P$ form a sphere).

Connected Concepts

• Locus of Points
• Intersecting Loci
• Perpendicular Lines
• Circle Fundamentals
• Analytic Geometry