What Is a Locus?

How a locus is formed

In classical geometry, locus language turns a condition into a shape. Points a fixed distance from one center form a circle, points equidistant from two fixed points form a perpendicular bisector, and points equidistant from two intersecting lines lie on angle bisectors.

A good locus argument always has two parts: every point on the proposed shape must satisfy the rule, and every point that satisfies the rule must lie on the proposed shape. That is why the topic matters in proof work as well as in drawing.

• A locus is the set of all points that satisfy a given condition.
• A locus describes every point that obeys the condition, not just a convenient example.
• Many familiar figures can be defined as loci, including circles, perpendicular bisectors, and conic sections.
• A complete locus explanation usually proves both inclusion directions: the rule leads to the shape and the shape fits the rule.

Where loci help

• Use locus when a geometry problem gives a distance, angle, or equidistance condition and asks for the shape that results.
• Use it in constructions, where the desired point is found by intersecting two different loci.
• Use it in analytic geometry to translate a geometric rule into an equation or family of equations.

Locus checks to make

• Do not describe only one point when the problem asks for the whole set of possible points.
• Do not assume a sketch is enough; the defining condition has to justify the entire shape.
• Do not stop after proving that points on the shape satisfy the rule if the reverse direction is still missing.