Is there a mathematical/geometric proof to show that, at most, 3 points will always lie in a common plane?

2023 © Wikiask
Main topic: Science
Other topics: Mathematics
Short answer:

There are three formulas that can show at most 3 points, which will always show they lie in a common plane. Only a line can be made from four collinear points. It is not possible to create a polygon. As a result, it is impossible to construct a quadrilateral using four collinear points.

  • Slope Formula
  • Area of Triangle Formula
  • Distance Formula

Collinear points are points that are located on the same line as another point. If there is no line on which all of the points can be found, then the points in question are said to be noncollinear.

When two distinct planes meet at an intersection, the result is always a line. If two planes that are parallel to one another are intersected by a plane, then the lines of intersection will also be parallel. Therefore, it is possible to create an endless number of planes that pass through three locations that are collinear with one another.

Collinear points.png

Formulas that show only 3 points can lie in a common plane?[edit]

Slope Formula

There are three points A, B, and C, and the three points are only going to be collinear if the slope of line AB is equal to the slope of line BC and the slope of line AC. We make use of slope formula to determine the slope of a line that connects two points.[1]

The gradient of the line that connects the points P(x1, y1) and Q(x2, y2) is:

m = (y2-y1) / (x2−x1)

Area of Triangle Formula

It is claimed that three points are collinear if the area of the triangle that is created by those three points is equal to zero. If three points are collinear, it becomes impossible for them to create a triangle because of the way the coordinate system works.[2]

(1/2) | [x1(y2 – y3) + x2(y3 – y1) + x3[y1 – y2]| = 0

Distance Formula

The distance can be determined between the first point and the second point by using the formula for distance, and then we determine the distance in between second point and the third point. After doing so, we determine whether or not the total of these two lengths is equivalent to the gap that exists between the first and third points. If the three points in question are collinear points, then this will be a viable option.[3]


  1. "Slope Formula - What is Slope Formula? Equation, Examples". Cuemath. Retrieved 2022-10-09.
  2. "Area of Triangle - Formula, How to Find Area of Triangle". Cuemath. Retrieved 2022-10-09.
  3. "Distance Formula - Derivation, Examples, Types, Applications". Cuemath. Retrieved 2022-10-09.