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CollinearPoints[{p1,p2,p3,…,pn}]

tests whether the points p1,p2,p3,…,pn are collinear.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Basic Applications  
Geometry  
Properties & Relations  
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CollinearPoints

CollinearPoints[{p1,p2,p3,…,pn}]

tests whether the points p1,p2,p3,…,pn are collinear.

Details

  • CollinearPoints is also known as rectilinear.
  • Typically used to test whether a set of points lie on the same straight line.
  • CollinearPoints[{p1,p2,p3,…,pn}] gives True if the points p3,…,pn are on the line passing through p1 and p2.
  • For collinear points p1, p2 and p3, the rank of the matrix {p2-p1,p3-p1} is less than or equal to 1.

Examples

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Basic Examples  (2)

The points {0,0},{1,2},{2,4} are collinear:

Wolfram Language code: pts = {{0, 0}, {1, 2}, {2, 4}};
Wolfram Language code: CollinearPoints[pts]

Plot the points:

Wolfram Language code: Graphics[{Line[pts], Red, Point[pts]}]

Find the equation of the line passing through the points {0,1} and {1,2}:

Wolfram Language code: CollinearPoints[{{0, 1}, {1, 2}, {x, y}}]

Scope  (4)

CollinearPoints works with two-dimensional points:

Wolfram Language code: CollinearPoints[{{1, 1}, {3, 4}, {5, 6}}]

Three-dimensional points:

Wolfram Language code: CollinearPoints[{{1, 2, 1}, {3, 4, 1}, {5, 6, 1}, {7, 8, 1}}]

-dimensional points:

Wolfram Language code: CollinearPoints[{{0, 0, 1, 1}, {1, 0, 1, 0}, {0, 1, 0, 1}, {0, 1, 1, 1}, {0, 0, 0, 1}}]

CollinearPoints works with numerical coordinates:

Wolfram Language code: CollinearPoints[{{1, 1}, {3, 4}, {5, 6}}]

Symbolic coordinates:

Wolfram Language code: CollinearPoints[{{0, 0, 1}, {1, 0, 1}, {0, 1, 1}, {a, b, c}}]

CollinearPoints over a set of coordinates:

Wolfram Language code: CollinearPoints[{{1, 2, 1}, {3, 4, 1}, {5, 6, 1}, {7, 8, 1}}]

List of points:

Wolfram Language code: CollinearPoints[{Point[{0, 0, 1}], Point[{1, 0, 1}], Point[{0, 1, 1}], Point[{a, b, c}]}]

Multi-points:

Wolfram Language code: CollinearPoints[Point[{{0, 0, 1}, {1, 0, 1}, {1, 3, 1}, {a, b, c}}]]

CollinearPoints works for large sets:

Wolfram Language code: pts = RandomPoint[Line[{{0, 0}, {1, 0}}], 10 ^ 6];
Wolfram Language code: CollinearPoints[pts]//AbsoluteTiming

Applications  (4)

Basic Applications  (3)

Find conditions for which two points lie on a line passing through the origin:

Wolfram Language code: CollinearPoints[{{0, 0}, {a, b}, {c, d}}]

Explicit instances:

Wolfram Language code: {{a, b}, {c, d}} /.  FindInstance[% && a ≠ 0, {a, b, c, d}, Reals, 3]

Find the equation of the line passing through the points {0,1} and {1,2}:

Wolfram Language code: CollinearPoints[{{0, 1}, {1, 2}, {x, y}}]

Draw collinear points:

Wolfram Language code: pts = {{0, 0}, {1, 1}, {2, 2}};
Wolfram Language code: CollinearPoints[pts]
Wolfram Language code: Graphics[{InfiniteLine[pts[[1 ;; 2]]], Red, Point[pts]}]

Geometry  (1)

Noncollinear points form a polygon:

Wolfram Language code: pts = {{0, 0}, {1, 3}, {2, 2}};
Wolfram Language code: CollinearPoints[pts]
Wolfram Language code: Graphics[Polygon[pts]]

Collinear points:

Wolfram Language code: pts = {{0, 0}, {1, 1}, {2, 2}};
Wolfram Language code: CollinearPoints[pts]
Wolfram Language code: Graphics[{EdgeForm[StandardGray], Polygon[pts]}]

Properties & Relations  (5)

PositivelyOrientedPoints returns False for collinear points:

Wolfram Language code: pts = {{0, 0}, {1, 2}, {2, 4}};
Wolfram Language code: {CollinearPoints[pts], PositivelyOrientedPoints[pts]}

NegativelyOrientedPoints returns False for collinear points:

Wolfram Language code: pts = {{0, 0}, {1, 2}, {2, 4}};
Wolfram Language code: {CollinearPoints[pts], NegativelyOrientedPoints[pts]}

Collinear points are coplanar:

Wolfram Language code: pts = RandomPoint[Line[{{0, 0, 0}, {1, 1, 1}}], 5];
Wolfram Language code: {CollinearPoints[pts], CoplanarPoints[pts]}

Use RegionMember to test whether points are collinear:

Wolfram Language code: RegionMember[Line[{{0, 0}, {1, 1}}], {2, 2}]
Wolfram Language code: CollinearPoints[{{0, 0}, {1, 1}, {2, 2}}]

Use InfiniteLine to draw a graphics image:

Wolfram Language code: pts = {{0, 0}, {1, 1}, {2, 2}};
Wolfram Language code: Graphics[{InfiniteLine[pts[[1 ;; 2]]], Red, Point[pts]}]
Wolfram Research (2020), CollinearPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/CollinearPoints.html.

Text

Wolfram Research (2020), CollinearPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/CollinearPoints.html.

CMS

Wolfram Language. 2020. "CollinearPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CollinearPoints.html.

APA

Wolfram Language. (2020). CollinearPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CollinearPoints.html

BibTeX

@misc{reference.wolfram_2026_collinearpoints, author="Wolfram Research", title="{CollinearPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/CollinearPoints.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_collinearpoints, organization={Wolfram Research}, title={CollinearPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/CollinearPoints.html}, note=[Accessed: 13-June-2026]}