# Center|Definition & Meaning

**Definition**

A **center** is generally **a point** at the **center of an object** in geometry. Centers **may or may not exist** depending on the definition of the term taken into consideration. The center is defined as the point **where all the isometries** that **move** the **object onto itself** have a **fixed point** if we **consider** **geometry** as the study of **isometry groups.**

**Circles, spheres, triangles**, and line segments **have** many **points** inside them, and each point is **situated** at a **different distance** from the other points. The **center of a geometric object** (circle, triangle, line segment) however, is comprised of **only one point**, which is always the** same distance** from **every other point**.

**Center of Circle**

Circle centers **correspond** to the **points** on a **circle** that are **equal distances** from **one another** in other words circle centers are **points** that are **equally distant** from their **edges**.

Consider the figure above where a **circle** is **drawn** the **point A** is **labeled** as the **center of the circle** where **AB** is the **radius** with **B** as a **point on** the **circle**. The reason we are calling A the center of the circle is that the **distance** of **all **the **points** in the circle from **point A**(center) is **equidistant.**

Consider the **right circle** in above figure **C** is **labeled** as the **Center** of the circle while **D, E, F, G, and H** are **points on** the **circle** so all of these points will be **equidistant** that’s what we define above the definition of a circle and referring C as the center of the circle.

**CD = CE = CF = CG = CH**

The **formula** for finding the center of circles can be expressed by the **general equation** for the circle. In a circle, consider **(L, M), r, and (X, Y)** as the **coordinates** of the** center** of the circle, the **radius** of the circle, and the **coordinates** of **any point** on the circle respectively. The center formula of the circle can be written as :

**(X – L)² + (Y – M)² =r²**

**Center of Sphere**

Spheres are **three-dimensional space** objects **formed** by **points** that are **equally distant** from a point and that point is known as the **center of the sphere**.

Consider the above figure **point A** is known as the **center** of the **sphere** because that **point** is **equidistant** from **all** the **points** on the **surface** of a sphere.

The **formula** for finding the sphere can be expressed by the **general equation** for the sphere. In a sphere, consider **(L, M, N), r, and (X, Y, Z)** as the **coordinates of the center** of the sphere, the **radius** of the sphere, and the **coordinates of any poin**t on the sphere respectively. The center formula of the sphere can be written as :

**(X – L)² + (Y – M)² + (Z – N)² =r²**

**Center of Triangle**

An actual **triangle center** or triangle center in **geometry** is a **point** on a **plane** that, in some sense, is in the **middle** of the **triangle**, similar to the **center** of a **square** or **circle.** There are many **distinct points** of **triangles** that are **referred** to as the **center** of the triangle as described below. Consider the below figure of an **equilateral triangle** the **center** is **labeled** as **H** where all **three vertices** of the triangle **pass**.

**Circumcenter**

** **Consider **three vertices** of the triangle, the **center** of the triangle **will be** the **point** that **passes** through all **three vertices**.

**Centroid**

** **Consider a **triangle** that has a **uniform density** so the **centroid** will be the** point** where the **triangle** is **balanced.**

**Incentre**

** **Consider a triangle the **incentre** can be defined as the **point** where the **internal angle bisector** **meets** or the center that is** tangent** to the **three sides** of the **triangle**.

**Orthocenter**

** **Consider a triangle the **orthocenter** can be **defined** as the point where the **altitudes** of the **triangle intersect**.

**Center of Polygon**

Given a regular polygon, the center of the polygon can be defined as a **point (k)** that is **equidistant** from **all** of the** vertices** as shown in the figure below.

There are other types of polygons like **cyclic and tangential polygons** we will also discuss their center as well.

**Center of Tangential Polygon**

Consider a **tangential polygon** that has all of its **vertices** **tangent** to a **distinct circle** that circle is known as an **incircle (inscribed circle)** and the center of that circle **(incenter)** is **referred** to as the** center** of the tangential polygon. The center of the tangential polygon is **A.**

**Center of Cyclic Polygon**

Consider a **cyclic polygon** that has **all** of its **vertices** on a **distinct circle** that circle is known as a **circumcircle** and the **center** of that circle **(circumcenter)** is known as the **center** of the **cyclic polygon** as shown in the figure. The center of the cyclic polygon is **A**.

**Center of Line Segment**

The Center of a **line segment** is usually referred to as the **midpoint** which is basically **mid of two points.** Let’s suppose two **points A and B** to find the midpoint we will calculate the **distance** between the **two points** and **divide** that **distance** into **two parts.**

For instance, there are two points and the **distance** between them **is 4cm** to find the **center** of the line segment i.e. the midpoint we will** divide** the **distance** by **2** and what we get is **2** so the **midpoint (center)** will on **2cm** from **A** and **2cm** from** B.** This concept is illustrated in the figure below. E is the midpoint of the line segment.

**Locating Centers – An Example**

Consider the following figure having two problems **A and B** the data is given in the figure we want to **find the center** of the** circle** and the center of the **line segment**.

Figure 8 – Example of the Center of a Circle and Line Segment

**Solution**

First, let us solve for **Figure A**.

There are two points given A and E, according to the given data

AB=**2cm**

AC = **2cm**

**Both** the **points** are **equidistant** while we are also given that:

EF = **3cm**

EG = **2cm**

EH = **4cm**

Three of the **given points** are **not** **equidistant**, so E is not the circle’s center. Thus, we conclude that **A is the center** of the circle as it is **equidistan**t from **all points** on the circle.

Moving on to **Figure B**, we have two **points** **K and L.**

To find the center, we will first **find** the **distance** between **I and J** which turns out to be **6cm. D****ividing** this distance **by 2** gives us **3cm**, so we will now **look** at the **point** that is **3cm away** from **I and J**. In the case of **L**, it is **4 cm away** from** I** and **2cm away** from **J** while in the case of **K,** it is **3cm away** from **I** and **3cm away** **from J** so we will conclude **K** is the **midpoint** or center of the **line segment**.

*All mathematical drawings and images were created with GeoGebra.*