Noncollinear

Noncollinear

Points that do not all lie on a single line.

See also

Worked Example

Problem: Determine whether the points A(1, 2), B(3, 4), and C(2, 5) are noncollinear.

Step 1: Find the slope between A and B.

m_{AB} = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1

Step 2: Find the slope between A and C.

m_{AC} = \frac{5 - 2}{2 - 1} = \frac{3}{1} = 3

Step 3: Compare the two slopes. Since the slope from A to B is 1 and the slope from A to C is 3, the slopes are not equal. This means C does not lie on the line through A and B.

m_{AB} \neq m_{AC} \implies \text{noncollinear}

Answer: The points A(1, 2), B(3, 4), and C(2, 5) are noncollinear because they do not all lie on the same line.

Another Example

Problem: Are the points P(0, 0), Q(4, 2), and R(8, 4) noncollinear?

Step 1: Find the slope between P and Q.

m_{PQ} = \frac{2 - 0}{4 - 0} = \frac{1}{2}

Step 2: Find the slope between P and R.

m_{PR} = \frac{4 - 0}{8 - 0} = \frac{1}{2}

Step 3: The slopes are equal, so all three points lie on the same line. They are collinear, not noncollinear.

m_{PQ} = m_{PR} = \frac{1}{2} \implies \text{collinear}

Answer: The points P, Q, and R are NOT noncollinear — they are collinear because all three lie on the line y = x/2.

Frequently Asked Questions

How many points do you need to be noncollinear?

You need at least three points. Any two points are always collinear because exactly one line passes through any two distinct points. Only with three or more points does the question of collinearity versus noncollinearity arise.

How do you prove three points are noncollinear?

The most common method is to compute slopes between pairs of points. If the slope from the first point to the second differs from the slope from the first point to the third, the points are noncollinear. Alternatively, you can use the area of the triangle formed by the three points: if the area is not zero, the points are noncollinear.

Noncollinear vs. Collinear

Collinear points all lie on a single straight line, while noncollinear points do not. Two distinct points are always collinear. Three or more points are collinear only when every point falls on the same line; if even one point falls off that line, the set is noncollinear. Noncollinear points can form shapes like triangles and polygons, whereas collinear points can only form line segments.

Why It Matters

Noncollinearity is a key requirement in many geometric constructions. For example, exactly one plane passes through any three noncollinear points — this is a foundational axiom in Euclidean geometry. You also need noncollinear points to define a triangle; three collinear points collapse into a line segment and form no triangle at all.

Common Mistakes

Mistake: Assuming two points can be noncollinear.

Correction: Any two distinct points always define exactly one line, so two points are always collinear. Noncollinearity only applies to sets of three or more points.

Mistake: Checking only one pair of slopes to conclude collinearity.

Correction: A single slope between two points tells you nothing about a third point. You must compare slopes across at least two pairs that share a common point (or use another method like the area test) to determine whether all points lie on the same line.

Related Terms

Collinear — Opposite concept — points all on one line
Point — The fundamental object being classified
Line — The reference object for collinearity
Coplanar — Points in the same plane — related spatial concept
Triangle — Requires three noncollinear points to form
Slope — Used to test if points are noncollinear