Incenter of a Triangle

A triangle is a type of three-sided two-dimensional polygon in geometry. The vertex of the triangle is formed when the two sides are united end to end. Between two sides, an angle is formed. Triangles have a variety of features, each of which can be studied at various levels of education. In this article, we will understand what is the incentive, the incentive of a triangle formula , incenter properties of the triangle and solved examples.

The point of the three interior angle bisectors of a triangle is the triangle's incenter. To put it another way, it's the place where the triangle's internal angle bisectors cross. As the point of the central axis is the center point of the triangle's inscribed circle, this location will be equidistant from the triangle's sides.

The center of a triangle's inscribed circle, which is the largest circle that will fit inside the triangle, is the incenter. This circle is also known as a triangle's incircle. Given below shows the incenter of the triangle.

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Incentre of a triangle formula

Let’s consider (x1, y1), (x2, y2) and (x3, y3) are the coordinates of vertices of a triangle ABC and a, b and c are the lengths of its sides, then the triangle’s incenter can be calculated using the incenter formula:

\[(\frac{ax_{1} + bx_{2} + cx_{3}}{a+b+c} , \frac{ay_{1} + by_{2} + cy_{3}}{a+b+c})\]

The formula above can be used to solve situations such as how to locate the incenter of a triangle using three coordinates.

After obtaining the lengths of sides of a triangle using the distance formula in coordinate geometry, we can simply substitute the coordinates in the formula to solve such problems.

Incentre of a triangle angle formula

Let E, F, and G represent the sites where C, A, and B's angle bisectors cross the sides AB, AC, and BC, respectively.

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We may compute the incenter of a triangle angle by using the angle sum attribute of a triangle.

In the above figure,

∠AIB = \[180^{\cdot } - \frac{(\angle A + \angle B)}{2}\]

Where I is the incenter of the given triangle

Incenter properties of a triangle

Following are the few important properties of triangles’ incenter.

• If I is the incenter of the triangle ABC (as indicated in the diagram), then the line segments AE and AG, CG and CF, BF and BE are all the same length, i.e. AE = AG, CG = CF and BF = BE.

• If I is the incenter of the triangle ABC, then we can say  ∠BAI = ∠CAI, ∠BCI = ∠ACI and ∠ABI = ∠CBI (using angle bisector theorem).

• We know that the sides of the triangle are tangents to the circle, hence , EI = FI = GI = r known as the inradii of the circle or radius of incircle.

• If s is the semi perimeter of the triangle and r is the inradius of the triangle, then the area of the triangle is equal to the product of s and r, i.e. A = sr.

• The incenter of a triangle is always within the triangle.

How to calculate incenter of a triangle

We have to locate the incenter of the triangles in two separate instances. By drawing the angle bisectors of the triangle, we may find the incenter during construction. In coordinate geometry, however, we can calculate the incenter using the formula.

Let’s understand these two concepts with the help of the below examples.

Example 1: Find the incenter coordinates of a triangle whose vertices are A(20, 15), B(0, 0), and C (-36, 15).

Solution: Let the given points be:

A(20, 15) = (x1,y1)

B(0, 0) = (x2,y2)

C(-36, 15) =(x3,y3)

The length of sides AB, BC, and CA can be calculated using the distance formula as follows:

\[AB = c = \sqrt{(20 - 0)^{2} + (15 - 0)^{2}} = \sqrt{625} = 25\]

\[BC = a = \sqrt{(0 - 36)^{2} + (0 - 15)^{2}} = \sqrt{1296 + 225} = 39\]

\[CA = b = \sqrt{(20 + 36)^{2} + (15 - 15)^{2}} = \sqrt{56^{2}} = 56\]

Substituting these values in incenter formula,

\[(\frac{ax_{1} + bx_{2} + cx_{3}}{a+b+c} , \frac{ay_{1} + by_{2} + cy_{3}}{a+b+c})\]

= [(780 + 0 -900)/(39 + 56 + 25), (585 + 0 + 375)/(39 + 56 + 25)]

= (-120/120, 960/120)

= (-1, 8)

Question 2. Draw an equilateral triangle with a side of 5 cm and its incircle using only a ruler and compasses.

Solution:

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Question 3. If I is the incenter of the triangle ABC then find the value of x in the figure.

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Solution: Given: I is the incenter of the triangle.

AI,BI and CI are the angle bisectors of the triangle, 

So, ∠BAI + ∠CBI + ∠ACI =\[\frac{180^{\cdot }}{2}\]

\[ 37^{\cdot} + 20^{\cdot} + x = 90^{\cdot} \]

\[57^{\cdot} + x = 90^{\cdot}\] 

x = \[90^{\cdot} - 57^{\cdot}\]

x=\[33^{\cdot }\]

Hence the value of x=\[33^{\cdot }\]

Incenter of a Right Triangle

The incenter of a right-angled triangle is given in the below diagram.

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Conclusion:

From the above article, we can conclude that one of the triangle's points of concurrency is the incentre, which is formed by the intersection of the triangle's three angle bisectors. The point of convergence is always found in the triangle's interior. It's the only point in the triangle where the sides' distances are equal. We have also discussed properties of the incentre of triangle.