Loci

A locus (plural: loci) is the complete set of all points — and only those points — that satisfy one or more given geometric conditions. Rather than thinking of a shape as a static figure, locus thinking asks: 'If I collect every point that obeys this rule, what path or region do I trace out?' The answer is almost always a familiar geometric object — a line, a circle, a parabola, or some combination — and discovering this connection is one of the most elegant ideas in Euclidean geometry. Loci appear throughout mathematics and its applications. The definition of a circle itself is a locus: the set of all points in a plane that are a fixed distance r from a fixed centre point. Similarly, the perpendicular bisector of a segment is the locus of all points equidistant from the two endpoints. Every conic section — circle, ellipse, parabola, hyperbola — can be defined as a locus satisfying a distance condition. Understanding loci therefore unlocks a unified way to think about curves and their equations. In this topic you will learn how to state locus conditions precisely, construct loci using compass-and-straightedge techniques, and describe them both geometrically and algebraically. You will also see how combining two locus conditions simultaneously (finding their intersection) lets you locate specific points — a powerful problem-solving tool used in navigation, engineering, and pure mathematics alike.

Key takeaways

State the condition precisely
Plot several trial points
Identify and construct the complete locus
Verify the locus
Find intersections of compound loci
Express the locus algebraically

State the condition precisely

Every locus problem begins with a clear condition. For example: 'all points at distance 3 cm from point A' or 'all points equidistant from lines l₁ and l₂.' Before drawing anything, write the condition in plain language and identify what types of distances or angles are being constrained.

Plot several trial points

Choose specific points that satisfy the condition and mark them. If a point must be 3 cm from A, use a compass set to 3 cm and mark points in several directions. These sample points give you a visual hint about what curve the full locus will form.

Identify and construct the complete locus

Recognise the pattern from your trial points and construct the full locus. For a fixed-distance condition from a point, the locus is a circle (compass centred at that point). For equidistance from two points, construct the perpendicular bisector of the segment joining them (bisect the segment, erect a perpendicular). For equidistance from two intersecting lines, bisect the angles between the lines.

Verify the locus

Check two things: (a) every point on the constructed figure satisfies the condition, and (b) every point satisfying the condition lies on the figure. This two-way check confirms completeness. For example, for the perpendicular bisector of AB, check that any point P on it satisfies PA = PB, and that any point Q with QA = QB must lie on that line.

Find intersections of compound loci

When a problem imposes two conditions simultaneously, construct each locus separately, then find their intersection points. For example, 'find all points 5 cm from centre O AND equidistant from lines l₁ and l₂' means: draw the circle of radius 5 centred at O, draw the angle bisectors of l₁ and l₂, and mark where they cross. Each intersection is a point satisfying both conditions.

Express the locus algebraically

Place a coordinate system and translate the geometric condition into an equation. The locus of points at distance r from point (a, b) becomes the equation (x − a)² + (y − b)² = r². The perpendicular bisector of the segment from (x₁, y₁) to (x₂, y₂) is the set of points where (x − x₁)² + (y − y₁)² = (x − x₂)² + (y − y₂)², which simplifies to a linear equation.

The Key Equation

(x - a)^2 + (y - b)^2 = r^2

x, y — Coordinates of a general point on the locus (length units): Any point P(x, y) that satisfies the equation lies exactly on the locus.
a, b — Coordinates of the fixed centre (length units): The fixed reference point from which the distance condition is measured.
r — Radius (length units): The constant distance from the centre to every point on the locus. It is the defining parameter of the circle.

Worked Example

Problem. Points A and B are 8 cm apart. Find the locus of all points P such that PA = PB, and describe it both geometrically and algebraically. Then find the specific points that also lie on the circle of radius 5 cm centred at A.

x = 4; \quad P_1 = (4,\,3), \quad P_2 = (4,\,-3)

Answer. The locus PA = PB is the perpendicular bisector of AB, which has equation x = 4. The points satisfying both conditions are (4, 3) and (4, −3).

Figure

Diagram

Origins · 1 minEuclid of Alexandria and how this was discovered

Euclid of Alexandria

-325–-265 · Greek (Ancient)

Euclid's Elements (c. 300 BCE) systematised all known geometry into 13 books built from five postulates and rigorous deductive proofs. The constructions in Book I — bisecting angles, erecting perpendiculars, constructing circles — are the exact tools used in locus problems today. His compass-and-straightedge framework defined what 'constructible' means for more than two thousand years. When King Ptolemy I asked Euclid if there was a shorter path to learning geometry than working through the Elements, Euclid reportedly replied: 'There is no royal road to geometry.' The phrase 'royal road' became a lasting metaphor for intellectual shortcuts that do not exist.

Perpendicular Bisector — The perpendicular bisector of a segment is one of the most important standard loci — the set of all points equidistant from two endpoints — and is constructed directly from the locus definition.
Angle Bisector — The angle bisector is the locus of all points equidistant from two intersecting lines, making it a key locus result used whenever angular distance conditions arise.
Circle: Definition and Properties — A circle is defined as the locus of points at a fixed distance from a centre — the most fundamental locus of all and the basis for the standard circle equation.
Conic Sections — Every conic section (ellipse, parabola, hyperbola) is defined as a locus satisfying focus-directrix distance conditions, extending locus ideas far beyond Euclidean construction.
Compass and Straightedge Constructions — All classical locus constructions — bisecting segments, erecting perpendiculars, drawing circles — rely on compass-and-straightedge techniques, making this an essential prerequisite skill.