Collinear vs. Coplanar Points
Are the points on the same line? On the same flat surface? These two vocabulary words describe the spatial relationship between any group of points.
Introduction
Some points share the same line; some share the same plane. Recognising these relationships is key for naming planes, spotting intersections, and writing proofs.
Past Knowledge
Points, lines, planes (1.1.1). Intersections (1.1.3).
Today's Goal
Identify collinear and coplanar points from diagrams and descriptions.
Future Success
Three non-collinear points define a plane — essential for naming planes and 3D reasoning.
Key Concepts
Collinear Points
Points that lie on the same line are collinear. Two points are always collinear (any two points determine a line). Three or more points may or may not be collinear.
Coplanar Points
Points that lie on the same plane are coplanar. Three points are always coplanar (any three non-collinear points determine a plane). Four or more points may or may not be coplanar.
Quick Check
Collinear points are automatically coplanar (a line lies in infinitely many planes). But coplanar points are not necessarily collinear.
Worked Examples
Identifying Collinear Points
Points A, B, and C all lie on line . Are they collinear?
Yes — by definition, points on the same line are collinear.
Answer: Yes, A, B, and C are collinear.
Non-Collinear but Coplanar
Points D, E, and F form a triangle. Are they collinear? Coplanar?
A triangle has three vertices not on the same line, so they are not collinear. Three points always determine a plane, so they are coplanar.
Answer: Not collinear, but coplanar.
Non-Coplanar Points
Three corners of a table and the tip of a light fixture above the table — are all four points coplanar?
The three table corners lie in one plane (the tabletop). The light fixture is above that plane, so the fourth point is not in the same plane.
Answer: No — the four points are not coplanar.
Common Pitfalls
Assuming 4 Points Are Always Coplanar
Three points are always coplanar, but a fourth point can be off the plane. Think of a tripod (3 legs = 1 plane) vs. adding a 4th leg that doesn't touch the ground.
Real-Life Applications
Tripods & Stability
A tripod with three legs always sits flat (3 points determine a plane). A four-legged table can wobble because the fourth leg might not be coplanar with the other three. That's why camera tripods use exactly three legs — guaranteed coplanarity.
Practice Quiz
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