Theorem

Points equidistant from two intersecting lines form a pair of bisectors

Geometry • Line • Angles at a Point

Statement

Two intersecting lines make four angles at their point of intersection. A point is equidistant from both lines exactly when it lies on one of the angle bisectors, and since vertically opposite angles share a bisector the four angles supply only two bisecting lines, which are perpendicular to each other. The locus is therefore a pair of perpendicular straight lines through the point of intersection, not a single line.

How to use it

1Take the two lines as the coordinate axes: the condition becomes $∣ x ∣ = ∣ y ∣$ , whose solution set is $y = x$ together with $y = - x$ .
2Inside a triangle only the internal bisectors are wanted, which is why the incentre is one point; the external bisectors give the excentres.
3If the two lines are parallel instead of intersecting, the locus collapses to the single line midway between them.

Worked example

Two lines meet at the origin along

y = 0

and

x = 0

. Find every point of the line

y = 3

that is equidistant from both lines.

Step-by-step

1
distance to $y = 0$ is $∣ y ∣ = 3$ ; distance to $x = 0$ is $∣ x ∣$
Why: Distance to an axis is the absolute value of the other coordinate.
2
$∣ x ∣ = 3$ , so $x = 3$ or $x = - 3$
Why: Equidistance gives two solutions, one on each bisector.
3
the points are $(3, 3)$ and $(- 3, 3)$

Answer

(3, 3)

and

(- 3, 3)

Watch out

Answering 'a straight line' is the standard error: the locus has two branches, because a point can sit in either of the two pairs of vertically opposite angles. Distances here are perpendicular distances to the lines, not to the point of intersection.

Builds on

Internal angle bisector of a triangle Angles on a straight line and vertically opposite angles

Also known as

locus equidistant from two lines, pair of angle bisectors, internal and external bisector, angle bisector locus, bisectors of two intersecting lines are perpendicular

Points equidistant from two intersecting lines form a pair of bisectors

Statement

Worked example

Appears in past papers

2020 (I)

2018 (I)