Incenter of a Triangle

Incentre is the center of the triangles where the bisectors of the interior angles intersect. The incentre is also called as the center of the triangle’s incircle. There are many properties that an incenter possesses. The incenter of a triangle means the triangle center, a point defined for any triangle in a way that is independent of the triangle’s placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of a triangle cross, as the point, is equidistant from the triangle’s sides, as the junction point of the medial axis and innermost point of the grassfire transform of that triangle, and as the center point of the inscribed circle of the triangle.

Definition of Incenter

The incenter of a triangle is the point of intersection of all three interior angle bisectors of the triangle. This point is equidistant from all the sides of a triangle, as the central axis’s junction point is the center point of the triangle’s inscribed circle. The incenter of a triangle is also known as the center of the triangle’s circle. The circle that is inscribed in a triangle is called the incircle of a triangle. The incenter is usually represented by the letter “I” The triangle ABC seen in the image below shows the incentre of a triangle.

Properties of an Incenter:

The incenter of a triangle has various properties, let us learn from the below image and state the properties one by one.

Property 1: If point I is the incenter of the triangle then line segments AE and AG, CG and CF, BF and BE are equal in length.

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

AI=AIAI=AI common in both triangles

IE=IGIE=IG radius of the circle

∠AEI=∠AGI=90∘∠AEI=∠AGI=90∘ angles

Hence △AEI≅△AGI △AEI≅△AGI

So, by CPCT side AE=AGAE=AG

Similarly, CG=CFCG=CF and BF=BEBF=BE.

Property 2: If point I is the incenter of the triangle, then ∠BAI=∠CAI∠BAI=∠CAI, ∠ABI=∠CBI∠ABI=∠CBI, and ∠BCI=∠ACI∠BCI=∠ACI.

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

We have already proved these two triangles congruent above.

So, by CPCT ∠BAI=∠CAI∠BAI=∠CAI.

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

Property 4: If s=(a+b+c)/2, where s is the semiperimeter of the triangle and r is the inradius of the triangle, then the area of the triangle will be: A = sr.

Property 5: Unlike the orthocenter, a triangle’s incenter always lies inside the triangle.

Incenter Formula

To calculate the incenter of a triangle with 3 coordinates, we have to use the incenter formula. Consider coordinates of incenter of the triangle ABC with coordinates of the vertices, A(x1,y1),B(x2,y2),C(x3,y3)A(x1,y1),B(x2,y2),C(x3,y3) and sides a,b,ca,b,c are:

Incenter of a Triangle Angle Formula:

To calculate the incenter of an angle of a triangle we can use the formula mentioned shown below:

Let E, F, and G be the points where the angle bisectors of C, A, and B cross sides AB, AC, and BC, respectively.

Using the angle sum property of a triangle, we will calculate the incenter of a triangle angle.

In the above figure,

∠AIB = 180° – (∠A + ∠B)/2

Where I is the incenter of the triangle.

How to Construct the Incenter of a Triangle?

The construction of the incenter of a triangle is possible with the help of a compass(rounder). The steps to construct the incenter of a triangle are given below:

• Step 1: Place one of the compass’s ends at one of the triangle’s vertices and the other side of the compass is on one side of the triangle.
• Step 2: Draw two arcs on two sides of the triangle(edge) using the compass.
• Step 3: By using the same width as before, draw two arcs inside the triangle so that they cross each other at a point from the point where each arc crosses the side.
• Step 4: Draw a line from the vertex of the triangle to the point where the two arcs inside the triangle meet.
• Step 5: Repeat the same process from the other vertex of the triangle too.
• Step 6: The point at which the two lines meet or intersect each other is the incenter of a triangle.

Conclusion:

Incentre is the center of the triangles where the bisectors of the interior angles intersect. The incentre is also called as the center of the triangle’s incircle. There are many properties that an incenter possesses. The incenter of a triangle means the triangle center, a point defined for any triangle in a way that is independent of the triangle’s placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of a triangle cross, as the point, is equidistant from the triangle’s sides, as the junction point of the medial axis and innermost point of the grassfire transform of that triangle, and as the center point of the inscribed circle of the triangle.