Home → Tutorials → Perpendicular Bisector Construction

Tutorial Beginner ⏱ 8 minutes

Perpendicular Bisector Construction

Use midpoints and perpendicular lines to locate geometry — essential for symmetric profiles, centre-finding, and equidistant point construction.

What You Will Build

A symmetric profile constructed using perpendicular bisectors — demonstrating how to find centres, create symmetric geometry, and locate equidistant points without manual calculation.

Operators Used

Point/Midpoint,L Line/P,Perpto,L Point/Intof,L,L
01

Create the Base Line

Create a diagonal line that will be bisected.

P0 = (10, 20) P1 = (70, 60) Line/P0, P1 → L0: from (10,20) to (70,60) Length = sqrt((70-10)^2 + (60-20)^2) = sqrt(3600+1600) = 72.111mm Slope = (60-20)/(70-10) = 40/60 = 2/3
02

Find the Midpoint

Create a point at the exact midpoint of L0.

Point/Midpoint, L0, 0 → P2 = (40, 40) [(10+70)/2, (20+60)/2] Verification: dist(P0, P2) = dist((10,20),(40,40)) = sqrt(900+400) = 36.056mm dist(P2, P1) = dist((40,40),(70,60)) = sqrt(900+400) = 36.056mm P2 is exactly equidistant from P0 and P1 ✓
03

Create the Perpendicular Bisector

Create a perpendicular line through the midpoint P2. This line is equidistant from P0 and P1 at every point.

Line/P2, Perpto, L0, Length=50, Dir=Left → L1: perpendicular bisector through (40,40) Direction: perpendicular to L0 (slope = -3/2) Extends 50mm upward-left from P2 Line/P2, Perpto, L0, Length=50, Dir=Right → L2: perpendicular bisector extending downward-right The full perpendicular bisector = L1 + L2 through P2.
Every point on the perpendicular bisector is exactly equidistant from P0 and P1. This is the geometric definition of a perpendicular bisector.
04

Use the Bisector to Find a Symmetric Point

Find a point on the perpendicular bisector at a specific distance from the midpoint.

P3 = point on L1 at distance 30mm from P2 Point/P2, Atangle, 90+angle(L0), Dist=30, 0 → P3: 30mm from P2 along the perpendicular direction Verification: dist(P3, P0) = dist(P3, P1) [equidistant from both endpoints] This is the circumcentre property.
05

Create a Circle Through Three Points

Use the perpendicular bisector to find the circumcentre — the centre of the circle passing through P0, P1, and a third point.

P4 = (40, 10) [third point] Line/P4, Perpto, Line/P0,P4, Length=40, Dir=Left → L3 [perpendicular bisector of P0-P4] Point/Intof, L1, L3 → P5 = circumcentre [intersection of two perpendicular bisectors] Circle/Centre, P5, R=dist(P5,P0), 0 → C0 [circumscribed circle through P0, P1, P4]
The circumcentre is equidistant from all three points. This is how Circle/3P works internally.

Tips & Best Practices

  • The perpendicular bisector of any line segment is equidistant from both endpoints.
  • Two perpendicular bisectors intersect at the circumcentre of the triangle formed by three points.
  • Use Dir=Left and Dir=Right to create both halves of the perpendicular bisector.
  • The midpoint operator (Point/Midpoint,L) is exact — no approximation.
  • This technique is used in Circle/3P internally to find the circumscribed circle centre.

Operators Used in This Tutorial

Point/Midpoint,L

Midpoint of Line

Creates a point at the midpoint of a line.
Line/P,Perpto,L

Perpendicular Line Segment

Creates a perpendicular segment from a point.
Point/Intof,L,L

Line Line Intersection

Creates a point at the intersection of two lines.
Circle/3P

Circle Through 3 Points

Creates a circle through three points.

Related Tutorials

Tutorial

Intersection-Based Construction

Use intersections to locate geometry precisely.
Tutorial

Rectangle with Filleted Corners

Apply the construction techniques to a practical profile.
Tutorial

Parallel Offset Profile

Create a profile offset from an existing profile.

← All Tutorials Operator Reference Try in XyzPilot →