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The **circumscribed** and **inscribed** circles of **triangles** play a crucial role in their properties. With their distinct positions and relationships to the triangle’s sides and angles, these circles offer fascinating insights into the nature of **triangles** and the interplay between their geometric elements.

In this article, we explore the captivating realms of the **circumscribed** and **inscribed** circles, uncovering their defining characteristics and the hidden secrets they unveil within the realm of **triangles**.

## Definition of Circumscribed and Inscribed Circles of Triangles

The **circumscribed** circle passes through all three vertices. It is a unique circle that encompasses the entire triangle within its circumference. The center of the **circumscribed** circle is equidistant from the three vertices of the **triangle**, and its radius is known as the **circumradius**.

On the other hand, the **inscribed** circle is a circle that is tangent to all three sides of the **triangle**. The **inscribed** circle lies entirely within the **triangle**, with its center coinciding with the intersection point of the angle bisectors of the **triangle**. The radius of the **inscribed** circle is referred to as the **inradius**.

The **circumscribed** and **inscribed** circles provide valuable geometric insights and properties of **triangles**, influencing various aspects such as angle relationships, side lengths, and perimeters. Exploring the characteristics and interplay between these circles sheds light on **triangles’** intrinsic geometry and symmetries.

Below we present a generic representation of **circumscribed and inscribed circles of triangles** in Figure-1.

Figure-1.

**Properties**

### Properties of the Circumscribed Circle:

#### Existence and Uniqueness

Every **non-degenerate triangle** (a triangle with **non-collinear** vertices) has a unique **circumscribed circle**.

#### Concurrency

The three **perpendicular bisectors** of the sides of a **triangle** intersect at a single point, the center of the **circumscribed** circle. This point is equidistant from the three vertices of the **triangle**.

#### Relationship with Angles

The angles subtended by the same arc on the **circumcircle** are equal. In other words, the measure of an **inscribed angle** is half the measure of the **central angle** intercepting the same arc.

#### Relationship with Sides

The length of a side of the triangle equals the diameter of the **circumscribed** circle multiplied by the sine of the angle opposite that side.

#### Circumradius

The radius of the **circumscribed** circle, known as the **circumradius**, can be calculated using the formula: **R = (abc) / (4Δ)**, where **a**, **b**, and **c** are the lengths of the triangle’s sides, and Δ represents the area of the triangle.

#### Maximum Circle

The **circumscribed circle** has the largest possible** radius** among all circles drawn around the **triangle**.

### Properties of the Inscribed Circle

#### Existence and Uniqueness

Every **non-degenerate** **triangle** has a unique** inscribed circle**.

#### Concurrency

The three **angle bisectors** of the **triangle** intersect at a single point, which is the center of the **inscribed** circle. This point is equidistant from the three sides of the **triangle**.

#### Relationship With Angles

The angles formed between the tangent lines from the **inscribed** circle’s center, and the **triangle’s** sides are equal.

#### Relationship With Sides

The radius of the **inscribed** circle, known as the **inradius**, can be calculated using the formula: **r = Δ / s**, where **Δ** represents the area of the triangle, and s is the semi-perimeter (half the sum of the lengths of the triangle’s sides).

#### Tangency

The **inscribed** circle is tangent to each side of the triangle at a single point. These points of tangency divide each side into two segments with lengths **proportional** to the **adjacent sides**.

#### Minimum Circle

The **inscribed** circle has the smallest possible radius among all circles that can be** inscribed** within the **triangle**.

**Applications **

### Trigonometry and Geometry

The properties of **circumscribed** and **inscribed** circles are fundamental to **trigonometric relationships** and **geometric constructions** involving **triangles**. They provide a basis for **angle measurements**, **side-length calculations**, and establishing **geometric proofs**.

### Surveying and Navigation

The **circumscribed circle** is applied in the **triangulation** process in **land surveying** and **navigation**. By measuring the angles and distances between known points, the position of an unknown point can be determined by constructing a **circumscribed circle** around the **triangle** formed by the known points.

### Architecture and Civil Engineering

The **circumscribed** and **inscribed circles** are essential in **architectural** and **civil engineering design**. For instance, in the construction of circular or polygonal buildings, the **circumscribed circle** helps determine the ideal size and shape of the structure. The **inscribed circle** aids in the placement of columns, pillars, or supports within a triangular layout.

### Circuits and Electronics

**Circumscribed** and **inscribed circles** are employed in circuit analysis and design in **electrical engineering**. For example, when constructing filters or resonant circuits, the properties of the **inscribed circle** are used to determine optimal component values and impedance matching.

### Computer Graphics and Animation

In computer graphics and animation, the **circumscribed** and **inscribed circles** play a role in rendering curved shapes and smooth animations. Algorithms that generate **curved surfaces** or **interpolate** points along a curve often utilize the properties of these circles to ensure accuracy and **smoothness**.

### Robotics and Kinematics

The **circumscribed** and **inscribed circles** are employed in **robotics** and **kinematics** for path planning and motion control. By using the properties of the **inscribed circle**, robots can navigate tight spaces and calculate optimal trajectories while **avoiding collisions**.

### Pattern Recognition and Image Processing

The properties of **circumscribed** and **inscribed circles** are utilized in **image processing** and **pattern recognition algorithms**. For instance, in shape recognition, these circles can be used as features to identify and classify objects based on their **enclosed shapes**.

**Exercise **

### Example 1

Given a triangle with side lengths **a = 5 cm**, **b = 7 cm**, and **c = 9 cm**, find the **circumradius (R)**.

### Solution

To find the circumradius, we can use the formula: **R = (abc) / (4Δ)**, where **Δ** represents the area of the triangle.

First, calculate the area of the triangle using** Heron’s** formula:

s = (a + b + c) / 2

= (5 + 7 + 9) / 2 = 10 Δ

Δ = √(s(s-a)(s-b)(s-c))

Δ = √(10(10-5)(10-7)(10-9))

Δ = √(10*5*3*1)

Δ = √150

Now, substitute the values into the formula:

R = (abc) / (4Δ)

R = (5 * 7 * 9) / (4 * √150)

R ≈ 6.28 cm

Therefore, the circumradius of the triangle is approximately **6.28 cm**.

Figure-2.

### Example 2

Finding the Inradius of a Triangle Given a triangle with side lengths **a = 8 cm**, **b = 10** cm, and **c = 12 cm**, find the **inradius (r).**

### Solution

To find the inradius, we can use the formula: **r = Δ / s**, where **Δ** represents the area of the triangle and s is the **semi-perimeter**.

First, calculate the area of the triangle using **Heron’s** formula:

s = (a + b + c) / 2

s = (8 + 10 + 12) / 2 = 15 Δ

Δ = √(s(s-a)(s-b)(s-c))

Δ = √(15(15-8)(15-10)(15-12))

Δ = √(15*7*5*3)

Δ = √1575

Now, substitute the values into the formula:

r = Δ / s

r = √1575 / 15

r ≈ 7.35 cm

Therefore, the inradius of the triangle is approximately **7.35 cm**.

Figure-3.

*All images were created with MATLAB.*