Triangle Centers

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Incenter of a Triangle

We all have spotted triangles or triangular objects in our day to day life. We are also familiar with the dimensions of a triangle that it has three sides and three angles, but what about other important elements of the triangle? We are talking about the incenter of a triangle. Incenter is actually the point of intersection of all the 3 interior angle bisectors of the triangle. On the other hand, the angle bisectors in a triangle are always concurrent.


Pictorial Presentation of Incenters of a Triangle

Let’s check out the simulation below in order to know the incenters of different triangles on the triangular field.

Paul has broken down all the 3 angles equally and stretched out the lines.

All the 3 lines bisect at one point.

So, Paul went to stand on that point.

Do you know what that point is called?

Yes, Paul is standing in the incenter on the field.

[Image will be Uploaded Soon]

 

Centroid of a Triangle Definition

The centroid of a triangle means the centre point of an object. The points in which the 3 medians of the triangle bisect are referred to as the centroid of a triangle. It is also described as the point of bisection of all the 3 medians. The median is a line which connects the center point of a side and the opposite vertex of the triangle. The centroid of the triangle divides the median in 2: 1. It can be simply calculated by taking the average of both x- coordinate points and y-coordinate points of all the vertices of the triangle. Hope the centroid of a triangle definition seems simple and clear with the given explanation.


Centroid of a Right Angle Triangle

The centroid of a right triangle is actually the point of intersection of 3 medians, constructed through the vertices of the triangle to the midpoint of the opposite sides.

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Types of Triangle

In Euclidean Geometry, the centroid is a crucial concept in reference to a triangle. A triangle is a 3-sided bounded figure consisting of interior angles. Depending upon the sides and angles, a triangle can be categorized into different types such as:

  • Equilateral triangle

  • Isosceles triangle

  • Scalene triangle

  • Acute-angled triangle

  • Obtuse-angled triangle

  • Right-angled triangle

The centroid is a crucial property of a triangle as well for different geometric shapes in detail.


Centroid of a Triangle Formula

Let’s assume a triangle ABC. If the 3 vertices of the triangle are A(x\[_{1}\], y\[_{1}\]), B(x\[_{2}\], y\[_{2}\]), C(x\[_{3}\], y\[_{3}\]), then the centroid of a triangle can be found out by taking the average of X and Y coordinate points of all 3 vertices. Thus, the centroid of a triangle can be mathematically written as:

Centroid of triangle formula = (\[\frac{x_{1}+x_{2}+x_{3})}{3}\]), (\[\frac{y_{1}+y_{2}+y_{3})}{3}\])


Solved Examples

Example:

Mahima computed the area of a triangular box as 180 square feet. The perimeter of the box is 60 feet. If a circle is inscribed in the interior of the triangle in a way that it touches every side of the triangle, help Mahima calculate the in radius of the triangle.


Solution:

Given measurements:

The area of the box = 180 square feet

The perimeter of the box = 60feet

Thus, the Semiperimeter of the triangular box = 60 feet =30 feet

The area of the triangle = sr where r is the inradius of the triangle

Area 

180 =30 × r

180/30 =r

6

=r

Therefore, r = 6 feet


Example:

The coordinates of the incenter of the triangle PQR formed by the points A(3,1),B(0,3),C(−3,1) is (p,q). Determine triangle points (p,q).


Solution:

Given:

The vertices of the triangles = A (3,1),B(0,3),C(−3,1)

c = AB = \[\sqrt{(3-0)^{2} + (1-3)^{2}}\]

c = AB = \[\sqrt{3^{2} + (-2)^{2}}\] = \[\sqrt{13}\]

a = BC = \[\sqrt{(-3-0)^{2} + (1-3)^{2}}\] 

a = BC = \[\sqrt{(-3)^{2} + (-2)^{2}}\] = \[\sqrt{13}\]

b = AC = \[\sqrt{(-3-3)^{2} + (1-1)^{2}}\]

b = AC = \[\sqrt{-6^{2}+0^{2}}\] = 6

Incenter of the triangle is:

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

(\[\frac{3\sqrt{13}+0−3\sqrt{13}}{6+2\sqrt{13}}\], \[\frac{2\sqrt{13}+18}{6+2\sqrt{13}}\])

(0, \[\frac{2\sqrt{13}+18}{6+2\sqrt{13}}\])

Therefore, (0, \[\frac{2\sqrt{13}+18}{6+2\sqrt{13}}\])


Fun Facts

  • The centroid of a triangle splits up the median in the ratio of 2:1.

  • The incenter of a triangle can also be described as the center of the circle which is stamped in a triangle

  • When a circle is inscribed in a triangle in a way that the circle touches each side of the triangle, the center of the circle is what we call the incenter of the triangle.

FAQ (Frequently Asked Questions)

Q1. What are the Properties of a Centroid of a Triangle?

Answer: Following are the properties of centroid:

  • The centroid is the centre of gravity.

  • It is the centre of the object and also the point of concurrency of the medians

  • It must always lie inside the object.

Q2. What are the Basic Properties of an Incenter of a Triangle?

Given below are the properties of an incenter:

  • Property 1: Given that the II is the incenter of the triangle, then line segments namely— AE and AG, CG and CF, BF and BE will be equivalent in length.

Proof: The triangles AEI and AGI are in congruence by the rule of RHS congruence.

AI=AI (common in both triangles)

IE=IG (radius of the circle)

∠AEI=∠AGI=90angles

Therefore, △AEI≅△AGI

Thus, by CPCT side AE=AG

In the same manner, CG=CF and BF=BE

  • Property 2: Given that the II is the incenter of the triangle, then angles ∠BAI=∠CAI, ∠ABI=∠CBI∠, and ∠BCI=∠ACI.

Proof: The triangles AEI and AGI are congruent triangles by RHS rule of congruence.

We have already proved that these two triangles are in congruence in the above proof.

Therefore, ∠BAI=∠CAI (by CPCT)

  • Property 3: The sides of the triangle are tangents to the circle, thus OE = OF = OG=r are known as the inradii of the circle.

  • Property 4: The coordinates of incenter of the triangle ABC with coordinates of the vertices, A(x1,y1),B(x2,y2),C(x3,y3) and sides a,b,c are:

(ax1+bx2+cx3 /a+b+c, ay1+by2+cy3 /a+b+c)

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