Contents

# Equidistant|Definition & Meaning

**Definition**

When the **distances between** a **point** and the **objects** in a set are **equal**, that **point** is said to be **equidistant** from the set. According to Euclidean geometry, the **perpendicular bisector connects two points** that are **equidistant** from each other. It follows that **when two points** are **equidistant** from **two other points** in **three dimensions**, the **point** is **located** in a **plane**.

**Overview **

Figure 1 – Midfielder Illustration for the equidistant concept

Suppose a **scenario** of a **football groun**d where many **players** have been **assigned** some **positions** as part of the **lineup**. **Take** the **example** of a **midfielder; the** **midfielder position** is **assigned** in the **middle** of the **ground** from the **goal post**. So we can say that the **midfielder** is an **equidistant** **point** that has to be **in center** from **both the goal post**. This concept is elaborated on in the above figure.

Figure 2 – Distance Between House and Apple Store for equidistant Concept

Suppose another scenario where your** house** is **located** near **two Apple stores**. The **distance** **from** one of the **apple stores** to your **house** is **5Km** and the **distance from** the **second Apple store** to your** house** is also **5 km**, but it is **not mandatory** that your** house will be in the middle** of both stores. Your h**ouse might be at perpendicular bisector** of both the stores. This scenario is elaborated on in the above figure.

**Equidistant in Triangle and Polygon**

Figure 3 – Circumcenter of Triangle Equidistant from its vertices

Typically, the **circumcenter** of a **triangle** is a **point** that is **equally distant from** each vertex. There is always a **point** at the c**enter of a non-degenerate triangle**.

Likewise, **cyclic polygons** have **equidistant circumcentres**: each **vertex** is **equally distant** from its** circumcentre**. **Triangles and other tangential polygons** have an** incentre equidistant** from **each** of their **points of tangency** with the circle.

Whenever a **perpendicular bisector** **connects two vertices** at the ends of a triangle or other polygon, **every point** is **equidistan**t from them. **Angles** have **bisectors** where the **points** that **emanate** from them are **equidistant** from each other.

Figure 4 – Center A of Regular Polygon Equidistant from its vertices

**Equidistant in Rectangle**

All **four vertices** of a **rectangle** are **equidistant** from the **center**, as are **two opposite sides**, and **two other opposite sides** are **equidistant** from the** center** as well.

**Equidistant in Circle and Sphere**

There is an** equidistant** **line** connecting the **center of a circle** to the **center of every point** on the circle, which **implies** that the** center** of the **circle** is **equidistant.** The same is the case for the **sphere** in which its **center** is** equidistant** from **each point** on the **surface**.

**Equidistant Parabola**

**Parabolas** are lines** perpendicular** to the **directrix** where **distances** between them are **determined** along a **line perpendicular** to the directrix, which is **equidistant** from the** focal point** and **directrix**, respectively.

**Equidistant Parallel Lines**

**Parallel lines** are defined as lines that are** infinite lines** and **never intersec**t each other. Consider the **two parallel lines** then, according to **euclidean geometry**, any **point** lying **on** the** first parallel line** is **equidistant** from the **point** lying **on** the **second parallel line** because the distance from a point, say p1, lying on line L1 to the point p2 lying on line L2 is same.

**Equidistant Points Formula**

Under the topic of equidistant, there are **two main formulas**. **Distance formula**s are used to **determine **the **distance between** any **two points**, and line segment **midpoint formulas** are used to **determine** line **segments’ midpoints**.

**Distance Formula**

The **length** of the **line connecting** two **points** is the **distance** between them. Distance between **two points** can be **obtained** by **subtracting** the **coordinates** that don’t match if the **horizontal or vertical line** is the **same**. Due to the fact that **Pythagorean distances** are calculated from **Cartesian coordinates,** they are **sometimes** called **Pythagorean distances**.Consider the points **p1, p2, p3 and p4**. We can write the **distance formula** for these points as follows:

d = $\sqrt{(p3-p1)^{2}+(p4-p2)^{2}}$

**Midpoint Formula**

Midpoints are the **points** in the **middle of line segments** in geometry. There is an **equal distance** between the **two endpoints**, **making** it the **centroid** of **both segments** and of the ends. Basically, it **cuts the segment** in **half.** Consider **coordinate pairs (a1,b1)** and **(a2,b2),** which represent the end of the line segments. We** can write** the **formula** for the **midpoint** as follows:

M = $\dfrac{(a1+a2)}{2},\dfrac{(b1+b2)}{2}$

**Properties of Equidistant**

**Two points**are considered**equidistant**if they have the**same distance**from a**specific location**which is usually said to be the midpoint.**Conversely,**the**midpoint**(center of the two points) is**equidistant****from that point**which shares an equal distance from the midpoint.- Generally, there are only
**two techniques**for**determining**if the given set of**points is equidistant:**the**Midpoint**and Euclidean**distance formula**.

**Steps for Checking Equidistant Criteria**

**Step 1**

In order to find whether the points are equidistant or not,** first** of all, **mark three things**.** Coordinates** of the **midpoin**t, coordinates of** two points** that we want to find whether they are **equidistant**.

**Step 2**

After finding the coordinates of two points and the coordinates of the midpoint **next step** is to **give the names** of **each coordinate**. For instance, for **point 1,** we can give **names x1 and y1**. For** point 2,** we can give **names x2 and y2** and for the **midpoint**, we can give names **Cx3 and Cy3.**

**Step 3**

**After naming** each coordinate, the **next step is to decide** **which formula** to **apply.** Either use** the distance formula** or the **midpoint formula** as described above. If we want to prove that the **center of a line segment** is **equidistant** from the **endpoints,** then we will **apply the midpoint formula,** and** if** we want to **prove that endpoints** are **equidistant** from the **midpoint** or any other point, we will **apply the distance formula**.

**Examples of Finding Distances Between Points**

**Example 1**

Consider a** line segment** **L1** having two points **(1,2) and (4,5).** **Find** the **distance** **between** the **points.**

**Solution**

(1, 2) = (p1, p2)

(4, 5) = (p3, p4)

d = $\sqrt{(p3-p1)^{2}+(p4-p2)^{2}}$

d = $\sqrt{(4-1)^{2}+(5-2)^{2}}$

d = $\sqrt{(3)^{2}+(3)^{2}}$

d = $\sqrt{(9+9)}$

d = $\sqrt{18}$

d = 4.242

**Example 2**

Consider the **endpoints** of the line segment as **(-5,4) and (1,-2)**. **Find** the **midpoint** that is** equidistant** from the **two points**.

**Solution**

(a1, b1) = (-5, 4)

(a2, b2) = (1, -2)

M = $\dfrac{(a1+a2)}{2},\dfrac{(b1+b2)}{2}$

M = $\dfrac{(-5+4)}{2},\dfrac{(4-2)}{2}$

M = $\dfrac{-1}{2},\dfrac{2}{2}$

M = (-0.5, 1)

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