JUMP TO TOPIC

Welcome to exploring the **fundamental principles** of **geometry**, focusing on the intriguing concept of **points** that lie on the same **line**.

In this article, we’ll dive deep into the **principles** of **lines** in **two-dimensional** and **three-dimensional** **spaces**, the tests for the **points** that lie on the same **line**, essential **theorems**, and the pervasive influence of these concepts in **modern geometry** and **real-world applications**.

**Definition**

In **geometry**, when **points** lie on the same **line**, they are said to be **collinear**. **Collinearity** is a property that relates **points** and **lines** in a geometric space. If two or more **points** fall on the same straight **line**, they are known as **collinear points**.

For instance, in a **two-dimensional** plane, if you draw a **straight line**, any number of **points** along that **line** would be considered **collinear**. This is because they all share the same **line** of reference. In **three dimensions**, the concept stays the same; if you can connect all the **points** using a single straight **line**, they are considered **collinear**. The word “**collinear**” comes from the Latin “**col**” (meaning “together”) and “**linear**” (meaning “line”), directly illustrating this concept.

This concept is a foundational aspect of many principles in **geometry** and is utilized in many **real-world applications**, from architectural design to **satellite navigation systems.**

Figure-1.

**Properties**

**Points that lie on the same line**, also known as **collinear points**, have several key properties. Here are some of the most notable properties:

**Unlimited Number of Points**

A **line** can have an **infinite number of points**. As long as these **points** are located on the same **straight line**, they are considered **collinear**.

**Two Points Form a Line**

Any **two points** in space will always form a **line**; thus, by definition, any two points are always **collinear**.

**Three or More Points**

When dealing with **three or more points**, **collinearity** becomes less trivial. **Three or more points are collinear** if they all lie on the same **straight line**.

**Measure of Collinearity**

We can determine if three points are **collinear** in a **two-dimensional** plane by examining the **slopes** between each pair of points. If the **slopes are equal**, the points are **collinear**.

In **three dimensions**, one common method is to calculate the **area of the triangle** formed by the points; if the **area is zero**, the points are **collinear**.

**Order of Points**

The order of points does not affect **collinearity**. For example, if **points A, B, and C are collinear**, it doesn’t matter whether B is between A and C or if C is between A and B. They remain **collinear**.

**Distance**

The **distance** between any two points is the **shortest** along the** line** on which the points lie.

**Line Segments**

When **three or more points are collinear**, the **line segment connecting the two extreme points** equals the **sum of the line segments connecting the intermediate points**.

**Exercise**

**Example 1**

Are the points **A(2,3), B(4,6),** and **C(6,9)** **collinear**?

### Solution

We can find the slopes between AB, BC, and AC. If they are equal, then the points are collinear.

Slope AB = (6-3)/(4-2) = 1.5

Slope BC = (9-6)/(6-4) = 1.5

Slope AC = (9-3)/(6-2) = 1.5

All three slopes are equal, so points **A**, **B**, and** C** are **collinear.**

Figure-2.

**Example 2**

Are the points **A(-1,-2), B(0,0),** and **C(3,6)** **collinear**?

### Solution

Again, we find the slopes:

Slope AB = (0-(-2))/ (0-(-1)) = 2

Slope BC = (6-0)/(3-0) = 2

Slope AC = (6-(-2))/(3-(-1)) = 2

Since the slopes are equal, the points **A, B,** and **C** are **collinear**.

**Example 3**

Are the points **A(2,4), B(5,7),** and **C(10,12)** **collinear**?

### Solution

We find the slopes:

Slope AB = (7-4)/(5-2) = 1

Slope BC = (12-7)/(10-5) = 1

Slope AC = (12-4)/(10-2) = 1

The slopes are equal, so the points **A, B,** and **C** are **collinear**.

**Example 4**

Are the points** A(-2,-3), B(-1,-2),** and **C(-3,-5)** **collinear**?

### Solution

We find the slopes:

Slope AB = (-2-(-3))/(-1-(-2)) = 1

Slope BC = (-5-(-2))/(-3-(-1)) = 1.5

Slope AC = (-5-(-3))/(-3-(-2)) = 2

The slopes are not equal, so points **A, B,** and **C** are **not collinear**.

Figure-3.

**ApplicationsĀ **

The concept of **points that lie on the same line**, or **collinearity**, is foundational to many areas of study and industry. Here are a few examples of its wide-ranging applications:

**Computer Graphics and Design**

In this field, the principles of **lines** and **points** are fundamental for creating visual content. The understanding of **points** and **lines** helps in creating **geometric shapes**, **digital illustrations**, **3D modeling**, and **animation**. **Collinearity** can be used in algorithms for **line drawing**, **polygon filling**, and more.

**Physics**

In physics, particularly in mechanics, the concept of **collinear points** is vital. For example, when analyzing motion in one dimension or forces acting along the same line, **collinearity** is a key consideration.

**Engineering**

In various branches of **engineering**, such as **civil**, **structural**, and **mechanical engineering**, the concept of **collinear points** is crucial. It helps in understanding **structural integrity**, **force distribution**, and **alignment** of different components of a machine or structure.

**Astronomy and Space Science**

**Collinearity** has important applications in **celestial navigation** and **satellite placement**. For example, **collinear points in space** are used to align telescopes with celestial bodies, or to **align satellites** in a **‘train’** to achieve constant coverage.

**Geography and Geometric Surveying**

In **geographic information systems (GIS)** and **surveying**, the concept of **collinearity** is used in identifying, representing, and manipulating **spatial relationships** between different** geographic phenomena**.

**Machine Learning and Computer Vision**

In the field of **machine learning** and particularly in **computer vision**, points lying on the same line (or more generally, geometric relationships between points) can be used for **object recognition**, **image stitching** for panorama creation, and **3D reconstruction** from multiple images.

**Robotics**

In robotics, **collinearity** plays a significant role in **motion planning** and **kinematics**. Robots use this basic principle to calculate trajectories and to optimize movements.

**Architecture and Urban Planning**

In **architecture** and **urban planning**, points that lie on the same line are used to ensure the proper alignment of structures. This alignment can be critical for **aesthetics, structural integrity, **and** functionality.**

*All images were created with GeoGebra.*