What is a locus?
A locus (plural loci) is the set of all points that obey a given rule. Think of a point that moves, but must always obey the rule. The path it draws is the locus.
To prove a figure is a locus, you show two things:
- Every point on the figure obeys the rule.
- Every point that obeys the rule is on the figure.
Both parts matter. A half-circle is not the locus of points 3 cm from O, because many good points are missing.
Four basic loci
1. Fixed distance from one point
All points at distance r from O form a circle with centre O and radius r. In space it is a sphere.
2. Equal distance from two points A and B
The locus is the perpendicular bisector of AB: the line through the midpoint of AB at 90° to it. Why? If PA = PB, triangles PMA and PMB (M = midpoint) are congruent by SSS, so PM is at 90° to AB.
3. Equal distance from two crossing lines
The locus is the pair of angle bisectors of the angles they make. Inside one angle it is just one bisector. Distance to a line is always measured along the perpendicular.
4. Fixed distance from a line
All points at distance d from a line form two parallel lines, one on each side, each d away.
Using two loci together: the circumcircle
A point that obeys two rules must lie on both loci, so it is where they cross.
Find a point equally far from A, B and C. Equal from A and B means it is on the perpendicular bisector of AB. Equal from B and C means it is on the perpendicular bisector of BC. These two lines meet at one point O. Then OA = OB = OC, so O is also on the third bisector. O is the circumcentre, and the circle with centre O and radius OA is the circumcircle.
Same idea: the angle bisectors of a triangle meet at the incentre, equally far from all three sides. It is the centre of the incircle.
Basic ruler-and-compass constructions (perpendicular bisector, angle bisector, perpendicular from a point) are all drawn using these loci.
The capable arc
Fix a segment AB. Find all points P on one side where angle APB equals a fixed angle α. The answer is an arc of a circle through A and B: the capable arc. This comes from the rule that angles in the same segment of a circle are equal.
How to draw it: draw the perpendicular bisector of AB. The centre C lies on it, with radius r = AB ÷ (2 sin α). If α = 90°, the arc is a semicircle on AB as diameter.
The full locus (both sides of AB) is two arcs, mirror images of each other.
Try it: the string and pin test
Put a pin in a sheet of card. Tie a 10 cm thread to it and a pencil at the other end. Keep the thread tight and move the pencil. You draw a circle: the locus of points 10 cm from the pin.
Now mark two dots A and B 8 cm apart. Use a ruler to find 6 points that are equally far from A and B. Join them. Do they make a straight line through the middle of AB?
Key formulas and definitions
- Fixed distance r from O → circle, centre O, radius r
- PA = PB → perpendicular bisector of AB
- Equal distance from two lines → angle bisectors
- Fixed distance d from a line → two parallel lines
- Capable arc radius: r = AB / (2 sin α)
- Circumcentre = meeting point of perpendicular bisectors
Worked examples
1. Describe the locus of points 5 cm from a point O.
Every such point is on a circle with centre O and radius 5 cm, and every point on that circle is 5 cm from O. So the locus is that circle.
2. A and B are 6 cm apart. Describe the locus of points equally far from A and B.
It is the perpendicular bisector of AB: the line through the midpoint M (3 cm from each end) at 90° to AB.
3. Two straight roads cross at 70°. A lamp must be equally far from both roads. Where can it go?
On either of the two angle bisectors. Inside the 70° angle the bisector makes 35° with each road. Inside the 110° angle it makes 55° with each road.
4. AB = 8 cm and O is its midpoint. Find the points 3 cm from O that are also equally far from A and B.
Locus 1 is a circle, centre O, radius 3. Locus 2 is the perpendicular bisector of AB, which passes through O. The line meets the circle at two points, 3 cm on each side of O. So there are 2 answers.
5. AB = 6 cm. Find the radius of the capable arc for 30°.
r = AB ÷ (2 sin 30°) = 6 ÷ (2 × 0.5) = 6 cm.
6. Triangle with A(0,0), B(6,0), C(0,8). Find the circumcentre.
Perpendicular bisector of AB: x = 3. Perpendicular bisector of AC: y = 4. They meet at (3, 4). Check: distance to A, B and C is √(9 + 16) = 5 each. Radius 5.
Common mistakes
- Drawing only part of the locus, for example one angle bisector when there are two.
- Measuring distance to a line along a slant instead of along the perpendicular.
- Forgetting the second check: every point that obeys the rule must be on the figure.
- Saying the locus of points at a fixed distance from a line is one line; it is two parallel lines.