Locus of Points

Definition

A locus is the set of all points, and only those points, that satisfy a specific geometric condition or set of conditions.

Why It Matters

The concept of a locus provides the geometric ‘DNA’ for complex shapes; understanding these underlying rules is necessary for everything from designing precise mechanical linkages to modeling the trajectories of celestial bodies.

Core Concepts

• The Dual Requirement: To prove a locus, one must show:
  1. Every point in the set satisfies the condition.
  2. Every point satisfying the condition is in the set.
• Fundamental Loci in a Plane:
  • Circle: Locus of points at a fixed distance ($d$) from a given point ($P$).
    • How to read: “The distance d.”
    • Meaning: Locus of points at fixed distance $d$ from point $P$—all such points form a circle of radius $d$.
  • Parallel Lines: Locus of points at a fixed distance ($d$) from a given line (a pair of lines).
  • Perpendicular Bisector: Locus of points equidistant from two fixed points ($P$ and $Q$).
  • Midway Line: Locus of points equidistant from two given parallel lines.
  • Angle Bisector: Locus of points equidistant from the sides of an angle.
  • Intersecting Line Bisectors: Locus of points equidistant from two intersecting lines (a pair of perpendicular lines bisecting the angles).
  • Concentric Circle: Locus of points equidistant from two concentric circles.
  • Concentric Pair: Locus of points at a fixed distance ($d$) from a circle with radius $r$. (If $r > d$, two circles; if $r < d$, one circle).
• Locus in Space:
  • Sphere: Locus of points at a fixed distance from a given point.
  • Plane: Locus of points equidistant from two fixed points (the perpendicular bisecting plane).

Connected Concepts

• Circle Fundamentals
• Analytic Geometry
• Conic Sections
• Geometric Proof Foundations