# How to check if four points are coplanar

## Introduction

The aim of this article is to check if four points are coplanar, i.e. they lie on the same plane. Let’s consider four points $$P_1$$, $$P_2$$, $$P_3$$ and $$P_4$$ defined in $$\mathbb{R}^3$$. This question may be reformulated as “is the point $$P_4$$ belongs to the plane defined by points $$P_1$$, $$P_2$$ and $$P_3$$“.

## Proof

First, let’s compute the normal vector to the plane defined by points $$P_1$$, $$P_2$$ and $$P_3$$:

$$\vec{n_1}=\vec{P_1P_2} \times \vec{P_1P_3}$$

Let’s now compute the normal vector to the plane defined by points $$P_1$$, $$P_2$$ and $$P_4$$:

$$\vec{n_2}=\vec{P_1P_2} \times \vec{P_1P_4}$$

If the points lie on the same plane, $$\vec{n_1}$$ and $$\vec{n_2}$$ are colinear and this can be check thanks to the dot product with this relation:

$$\vec{n_1} \cdot \vec{n_2} =0$$

This can be rewriten:

$$(\vec{P_1P_2} \times \vec{P_1P_3}) \cdot (\vec{P_1P_3} \times \vec{P_1P_4}) = 0$$

The previous equation can be simplified:

$$\vec{P_1P_2} \cdot (\vec{P_1P_2} \times \vec{P_1P_4}) =0$$

## Conclusion

The four points are coplanar if, and only if $$\vec{P_1P_2} \cdot (\vec{P_1P_3} \times \vec{P_1P_4}) =0$$.