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In geometry, we define a triangle as a type of 2D polygon, which has three sides. When the two sides are connected end to end, it is named the vertex of the triangle. This joining of sides forms an angle between the two sides. Through this article on the incenter of a triangle, we will aim to learn the definition, incentre formula, calculation and more.

Triangles hold different properties, and knowing each of these properties of triangle are important to have complete knowledge about triangles. One such important property is the incenter of a triangle.

The incenter is one of the centers of the triangles which is the point where the bisectors of the interior angles meet. The incentre is also known as the center of a triangle’s incircle. Let us learn more about the same.

The incenter of a triangle denotes the intersection point of all the three interior angle bisectors of the given triangle. In other words, we can see that the point where the internal angle bisectors of the triangle cross are said to be the incentre of a triangle.

This location will be at an equal distance from the sides of a triangle, as the central axis’s junction point is the centre mark of the triangle’s inscribed circle. The incenter of a triangle is also acknowledged as the center of a triangle’s circle as the largest circle could implement inside a triangle.

The circle that is inscribed in a triangle is named the incircle of a triangle. The incenter is usually denoted by the letter I. The triangle ABC as can be seen in the below image presents the incentre of a triangle.

Learn more about Area of a Triangle.

All triangles possess an incenter, and it regularly lies inside the triangle. One of the approaches to obtain the incenter is by applying the property that the incenter is the junction of the three angle bisectors, relating coordinate geometry to determine the incenter’s position. But this process is often computationally tiresome.

Generally, the simplest way to get the incenter is by first determining the inradius, or radius of the incircle, which is generally denoted by the letter “r”. Once the inradius is known to us, each side of the triangle can be explained by the length of the inradius, and the intersection of the resulting three lines will be the incenter. This approach again can be performed using coordinate geometry. Alternatively, the below-discussed formula can be used.

To obtain the incentre of a triangle formula assume \((x_1, y_1)\), \((x_2, y_2)\) and \((x_3, y_3)\) as the three coordinates of vertices of a triangle ABC respectively. Where a, b and c denote the lengths of its sides, then the triangle’s incenter can be calculated applying the formula as shown:

The coordinates of incentre are:

\(\left(\frac{ax_1+bx_2+cx_3}{a+b+c},\frac{ay_1+by_2+cy_3}{a+b+c}\right)\)

The above formula supports solving the questions like how to calculate the incenter of a triangle with three coordinates. To answer such problems, we can just substitute the coordinates in the formula after determining the lengths of sides of a triangle applying the distance formula in coordinate geometry.

To estimate the incenter of an angle of a triangle we can practice the formula introduced as follows:

Assign E, F and G to be the points where the angle bisectors of C, A and B intersect the sides AB, AC and BC, respectively. Applying the angle sum property of a triangle, we can determine the incenter of a triangle angle.

In the above figure,

∠AIB = 180° – (∠A + ∠B)/2

Where “I” implies the incenter of the provided triangle.

Also, read about Centroid of a Triangle.

With the knowledge of definition and formula let us learn some of the important properties of the incentre of a triangle.

Property 1: If ‘I’ signifies the incenter of the triangle ABC, then line segments AE and AG, CG and CF, BF and BE are identical in length, i.e. AE = AG, CG = CF plus BF = BE.

Property 2: Consider the same above figure, if ‘I’ implies the incenter of the triangle, then ∠BAI=∠CAI, ∠ABI=∠CBI, and ∠BCI=∠ACI (applying angle bisector theorem).

Property 3: The sides of the triangle signify tangents to the circle, and therefore, EI = FI = GI = r is identified as the inradii of the circle or radius of incircle.

Property 4: If\(s=\frac{\left(a+b+c\right)}{2}\), where ‘s’ denotes the semi perimeter of the triangle and ‘r’ stands for the inradius of the triangle, then the area of the triangle can be determined with the formula: A = sr.

Property 5: In contrast to the orthocenter, a triangle’s incenter constantly rests inside the triangle.

Learn more about Geometric Shapes here.

To understand how to find the incenter of a triangle we must have a basic idea of the construction of the incenter of a triangle. Below are some of the quick steps to construct the incenter of a triangle:

Know more about Height and Distance here.

To know more about how to find the incenter of a triangle let us practical some solved examples:

Question 1: In Triangle PQR ∠PQR = 35° and ∠PRQ = 45°, If I is the incenter of a triangle PQR then the value of ∠QIR is?

Solution:

Let us depict the given data on a diagram.

Given in the Triangle PQR, ∠PQR = 35 and ∠PRQ = 45

Since I is the incenter, angular bisector QI bisects ∠PQR, ∴ ∠IQR = 17.5

Similarly, IR is the angular bisector for ∠PRQ, ∴ ∠IRQ = 22.5

Since in a triangle sum of angles is 180 degrees, we have ∠IQR + ∠IRQ + ∠QIR = 180

17.5 + 22.5 + ∠QIR = 180

∴ ∠QIR = 140 degrees.

Example 2: Rajan determined the area of a triangular sheet as 80 feet square. The perimeter of the sheet is 20 feet. If a circle is drawn inside the triangle in such a way that it is meeting every side of the triangle, then calculate the inradius of the triangle.

Solution:

Given:

The area of the sheet = 80 feet square.

The perimeter of the sheet = 20 feet.

Semiperimeter of the triangular sheet =20 feet/2 = 10 feet.

The area of the triangle is given by the formula= sr, where r is the inradius of the triangle.

Area = sr

80 = 10 × r

r = 80/10

r = 8

Hence, r =8 feet.

Know more about Mean Median Mode here.

Example 3: Obtain the coordinates of the incenter of the triangle whose vertices are A(3, 2), B(0, 2) and C(-3, 2).

Solution :

The vertices of the triangle are A(3, 2), B(0, 2) and C(-3, 2).

Let ‘a’ be the measure of the side opposite to the vertex A, ‘b’ be the measure of the side opposite to the vertex B and ‘c’ be the measure of the side opposite to the vertex C.

That is,

AB = c, BC = a and CA = b

Using the distance formula to obtain the values of ‘a’, ‘b’ and ‘c’.

a = BC = \(\sqrt{[(0+3)^{2} + (2-2)^{2}]} = \sqrt{9} = 3\)

b = AC = \(\sqrt{[(3+3)^{2} + (2-2)^{2}]} = \sqrt{36} = 6\)

c = AB = \(\sqrt{[(3-0)^{2} + (2-2)^{2}]} = \sqrt{9} = 3\)

Incenter I, of the triangle is given by the formula:

\(\left(\frac{ax_1+bx_2+cx_3}{a+b+c},\frac{ay_1+by_2+cy_3}{a+b+c}\right)\)

Here,

\((x_1, y_1)\) = (3, 2)

\((x_2, y_2)\) = (0, 2)

\((x_3, y_3)\) = (-3, 2)

a = 3, b = 6 and c = 3

Then,

\(ax_1 + bx_2 + cx_3 = 3(3) + 6(0) + 3(-3) = 0\)

\(ay_1 + by_2 + cy_3 = 3(2) + 6(2) + 3(2) = 24\)

a + b + c = 3 + 6 + 3 = 12

Substituting the above values in the formula.

Incenter is = \(\left(\frac{0}{24},\frac{12}{24}\right)\).

= (0, 0.5)

To wind the topic let us look at some of the other relevant triangle properties and terms:

The area is defined as the region involved inside the boundary of an object/figure. The measurement is carried out in square units with the standard unit being square meters

\(m^2\). The area of a triangle is equivalent to half of the base times height. i.e.

Area of a Triangle (A)=\(\frac{1}{2}\times b\left(base\right)\times h\left(height\right)\).

The formula to find the centroid of a triangle is given by:

\(Centroid=C= \frac{\left(x_1+x_2+x_3\right)}{3}, \frac{\left(y_1+y_2+y_3\right)}{3}\)

Check more topics of Mathematics here.

Terms | Definition |

Incenter: | The incenter is one of the centers of the triangles which is the point where the bisectors of the interior angles meet. |

Circumcenter: | The circumcenter of the triangle is the point of junction of the three perpendicular bisectors. The perpendicular bisector of a triangle is the lines drawn perpendicularly from the midpoint of the triangle. |

Centroid: | The term centroid is defined as the centre point of the object. The point at which the three medians of the triangle intersect or touch each other is recognised as the centroid of a triangle. |

Orthocenter: | The orthocenter is the location where the three altitudes of a triangle meet. A line segment developed from one vertex to the opposite side, which is perpendicular to the opposite side, is known as the altitude of a triangle. |

We hope that the above article on Incenter of a Triangle is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

If you are checking Incenter of a Triangle article, also check the related maths articles in the table below: | |

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