Locus in Geometry: The Shape of a Rule

Learn to treat a locus as all points that fit a condition, recognize the big four loci, and solve harder problems by intersecting simpler ones. You will also learn quick checks that catch most mistakes fast.

A locus in geometry is what you get when a rule chooses points, and you collect every point that passes the rule. That sounds abstract until you notice something. Rules do not just pick scattered dots. Many rules force points to line up into familiar shapes. A circle is a rule. A straight line can be a rule. Even a perpendicular bisector is a rule. Once you start seeing shapes as rules made visible, locus questions stop feeling like puzzles and start feeling like translation.

Locus means all points that satisfy

A locus is the set of all points that satisfy a condition. The condition is usually about distance, being equally far from things, or staying on a boundary. The shape is not something you guess. It is what the rule allows.

A useful mental model is a paintbrush that can only move in certain ways. Wherever it is allowed to go, it leaves ink. The ink trail is the locus.

Before you read examples, it helps to watch a point obey a rule and leave its trail.

When the trail looks like a circle, the hidden rule is usually fixed distance from a point. When it looks like a straight line, the rule is often fixed distance from a line or equal distances from two points.

Rule to shape
When a condition involves one fixed object, the locus often wraps around it. When it involves two objects, the locus often sits in between them.

Two classic loci you should recognize fast

Fixed distance from a point gives a circle

If a point $P$ must stay exactly $r$ units from a fixed point $A$, then $AP=r$. Every point that satisfies that condition forms a circle with center $A$ and radius $r$. Not inside it, not outside it. Exactly on the boundary.

Fixed distance from a line gives two parallel lines

If a point $P$ must stay exactly $d$ units from a line $l$, the locus is not one line. It is two lines parallel to $l$, one on each side, each at perpendicular distance $d$ from $l$. Distance to a line means the shortest distance, measured perpendicularly.

To build intuition, change the distance and watch the locus shift while the rule stays the same.

A circle grows smoothly as $r$ increases. Parallel lines slide outward as $d$ increases. Same idea, different anchor.

Equidistant from two points gives the perpendicular bisector

The locus of points equidistant from two points $A$ and $B$ is the perpendicular bisector of segment $AB$. Every point on that line has equal distance to $A$ and $B$, and every point with equal distances to $A$ and $B$ lies on that line.

Why it works has a simple core. If $P$ is equally far from $A$ and $B$, then triangle $PAB$ has two equal sides $PA$ and $PB$. That forces $P$ to sit on the symmetric line that cuts $AB$ into two equal halves at a right angle.

Seeing the equal distances update as you move $P$ makes the logic feel inevitable.

One quick check. If your point is closer to $A$ than to $B$, it cannot be on the perpendicular bisector. The bisector is the exact fence where closeness swaps.

Equidistant from two intersecting lines gives angle bisectors

Distance to a line means perpendicular distance. So being equidistant from two intersecting lines means your perpendicular drop to each line has the same length.

That condition produces two loci.

• The internal angle bisector, which runs through the angle between the lines.
• The external angle bisector, which runs through the opposite angle outside.

Both are valid because there are two ways to be equally close to both lines.

Use the comparison to notice which bisector your point follows as it stays equally far from the two lines.

Two bisectors
If the lines intersect, equidistant points form two straight lines through the intersection, not one curve.

How to solve locus problems by intersecting rules

Most exam style locus problems stack conditions. Each condition produces a simple locus. The answer is where those loci overlap. That overlap is their intersection.

A reliable workflow looks like this.

• Convert each phrase into a distance or equal distance statement.
• Draw the locus for each statement separately.
• Intersect the loci, because points must satisfy all conditions at once.

For example, 2 cm from $A$ is a circle centered at $A$ with radius 2 cm. 3 cm from a line $l$ is two lines parallel to $l$ at distance 3 cm. The points that satisfy both are where the circle crosses those parallel lines. That could be two points, one point if it is tangent, or no points.

Try building a combined locus and focus on the overlap, not the full shapes.

A good habit is to say out loud what intersection means. Every answer point must pass both rules, like a filter that only keeps points that survive every test.

Common pitfalls and quick checks

Some mistakes come from tiny wording details that change the locus completely.

Distance from a line is always perpendicular. If you measure along a slant, you are not measuring distance to the line.

Segments versus lines matters. If the condition mentions segment $AB$, then points might be restricted to a region near that segment, not the infinite line through it. If it mentions line $AB$, it extends forever.

Boundary cases are also part of the game.

• Empty locus when conditions cannot be met together.
• Single point when two loci touch at exactly one point.
• Two points when two curves cross twice.

Use the toggles to see common wrong turns and the correction that fixes them.

One fast check works across many problems. Pick an easy test point you can reason about, like a midpoint, a point on a line, or the intersection of two lines. If it should satisfy the rule but your drawn locus misses it, something is off.

What’s next

Loci are the bridge between geometry pictures and algebra. In coordinate geometry, a locus becomes an equation. The circle rule $AP=r$ becomes $(x-a)^2+(y-b)^2=r^2$. The perpendicular bisector becomes a linear equation you can derive by setting distances equal.

Outside math class, the same idea appears as coverage regions. A phone tower with range $r$ creates a circular boundary. A robot that must stay a fixed distance from a wall traces parallel paths. Locus is the habit of asking what shape a rule forces, before you compute anything.