Mathematics.

circles

Loci and Constructions

Geometry30 minDifficulty4 out of 10

You should know: circles

Overview

A locus (plural: loci) is the set of all points satisfying a given geometric condition — for example, 'all points a fixed distance from a center' defines a circle, and 'all points equidistant from two fixed points' defines the perpendicular bisector of the segment joining them. A compass-and-straightedge construction is a method for drawing exact geometric figures using only an unmarked straightedge (for drawing lines through two points) and a compass (for drawing circles/arcs of a given radius from a given center), following the rules laid out by Euclid over two thousand years ago. Constructions are really loci made physical: bisecting a segment, for instance, is done by intersecting two circular loci centered at its endpoints.

Intuition

Think of a locus as the answer to 'where can I stand so that ___ is true?' If the condition is 'exactly 5 units from point P,' the answer is a circle of radius 5 around P. If the condition is 'the same distance from points P and Q,' the answer is a line — the perpendicular bisector of PQ — because it's the boundary between points closer to P and points closer to Q. Constructions exploit exactly this: to construct the perpendicular bisector of a segment, draw two circular arcs of equal radius (greater than half the segment) centered at each endpoint; the two arcs' intersection points are, by definition, equidistant from both endpoints, so the line through them IS the perpendicular bisector locus.

Interactive Graph

Construct a perpendicular bisector and angle bisector step by step with compass-and-straightedge moves

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Formal Definition

Definition

For fixed points P, Q and fixed lines ℓ₁, ℓ₂, the standard loci and their construction methods are:

{X:dist(X,P)=r}=circle of radius r centered at P\{X : \text{dist}(X,P) = r\} = \text{circle of radius } r \text{ centered at } P
Locus of points at fixed distance from a point (a circle)
{X:dist(X,P)=dist(X,Q)}=perpendicular bisector of PQ\{X : \text{dist}(X,P) = \text{dist}(X,Q)\} = \text{perpendicular bisector of } PQ
Locus of points equidistant from two fixed points
{X:dist(X,ℓ1)=dist(X,ℓ2)}=angle bisector of the angle between ℓ1,ℓ2\{X : \text{dist}(X,\ell_1) = \text{dist}(X,\ell_2)\} = \text{angle bisector of the angle between } \ell_1, \ell_2
Locus of points equidistant from two intersecting lines
{X:dist(X,ℓ)=d}=two lines parallel to ℓ, distance d on either side\{X : \text{dist}(X,\ell) = d\} = \text{two lines parallel to } \ell,\ \text{distance } d \text{ on either side}
Locus of points at fixed distance from a line

Notation

NotationMeaning
dist(X,P)\text{dist}(X,P)The distance between point X and point P
⊥-bisector\perp\text{-bisector}The perpendicular bisector: the line through the midpoint of a segment, perpendicular to it

Properties

Circle as a locus

The locus of points at fixed distance r from a center P is precisely the circle of radius r centered at P.\text{The locus of points at fixed distance r from a center P is precisely the circle of radius r centered at P.}

Perpendicular bisector as a locus

The locus of points equidistant from two fixed points P and Q is the perpendicular bisector of segment PQ.\text{The locus of points equidistant from two fixed points P and Q is the perpendicular bisector of segment PQ.}

Angle bisector as a locus

The locus of points equidistant from two intersecting lines is the angle bisector of the angle they form.\text{The locus of points equidistant from two intersecting lines is the angle bisector of the angle they form.}

Allowed construction moves

Only two tools: an unmarked straightedge (draw the line through two given points) and a compass (draw a circle of a given radius from a given center).\text{Only two tools: an unmarked straightedge (draw the line through two given points) and a compass (draw a circle of a given radius from a given center).}

Classic constructible operations

Bisecting a segment or angle, erecting a perpendicular, copying an angle, and inscribing regular polygons (e.g. equilateral triangle, square, regular hexagon, regular pentagon) are all classically constructible.\text{Bisecting a segment or angle, erecting a perpendicular, copying an angle, and inscribing regular polygons (e.g. equilateral triangle, square, regular hexagon, regular pentagon) are all classically constructible.}

Applications

CAD software constraint solvers encode the same perpendicular-bisector and angle-bisector loci to let designers specify 'equidistant' or 'symmetric' relationships precisely.

Worked Examples

  1. The set of points at a fixed distance from a single center point is, by definition, a circle.

    {X:dist(X,O)=4}=circle of radius 4 centered at O\{X : \text{dist}(X,O) = 4\} = \text{circle of radius } 4 \text{ centered at } O

Answer: A circle of radius 4 centered at O.

Practice Problems

Difficulty 2/10

Describe the locus of points equidistant from two fixed points A and B.

Difficulty 3/10

A goat is tethered by an 8 m rope to a stake in an open field. Find the area of the region the goat can graze.

Difficulty 5/10

Two straight roads cross at a point and a company wants to build a warehouse equidistant from both roads. Describe the locus of valid warehouse locations, and explain how to construct it with compass and straightedge.

Common Mistakes

Common Mistake

Thinking a locus is a single point rather than a whole set of points.

A locus is always the complete collection of ALL points satisfying the condition — e.g. 'equidistant from two points' describes an entire line (the perpendicular bisector), not just one point.

Common Mistake

Believing a compass-and-straightedge construction can use a marked ruler to measure arbitrary lengths.

Classical constructions restrict the straightedge to being unmarked (it may only draw a line through two already-known points) and the compass to drawing circles from already-known centers and radii — no direct length measurement is allowed.

Quiz

The locus of points equidistant from two fixed points P and Q is:
Which tools are allowed in a classical compass-and-straightedge construction?
The locus of points at a fixed distance from a single point P is:

Summary

  • A locus is the complete set of points satisfying a given geometric condition.
  • Fixed distance from a point → a circle; equidistant from two points → the perpendicular bisector; equidistant from two lines → the angle bisector.
  • Compass-and-straightedge constructions use only an unmarked straightedge and a compass, following Euclid's classical rules.
  • The perpendicular bisector construction (equal-radius arcs from each endpoint) works because both arc intersection points satisfy the equidistant locus condition.
  • Loci and constructions are the same idea from two angles: a locus describes a set abstractly; a construction builds it physically with two tools.

References

  1. WebsiteWikipedia — Locus (mathematics)
  2. WebsiteWikipedia — Compass-and-straightedge construction