Incenter of a Triangle
Incentre is one of the centers of the triangles where the bisectors of the interior angles intersect. The incentre is also called the center of a triangle's incircle. There are different kinds of properties that an incenter possesses. In this section, we will learn about the incenter of a triangle by understanding the properties of the incenter, the construction of the incenter, and how to apply them while solving problems.
1.  Definition of Incenter 
2.  Properties of an Incenter 
3.  Incenter Formula 
4.  Incenter of a Triangle Angle Formula 
5.  FAQs on Incenter 
Definition of Incenter
The incenter of a triangle is the point of intersection of all the three interior angle bisectors of the triangle. This point is equidistant from 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 a triangle's circle since the largest circle could fit inside a triangle. The circle that is inscribed in a triangle is called an 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 look at the below image and state the properties onebyone.
Property 1: If 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 \(\text{AEI}\) and \(\text{AGI}\) are congruent triangles by RHS rule of congruency.
\(\text{AI} = \text{AI}\) common in both triangles
\(\text{IE} = \text{IG}\) radius of the circle
\(\angle \text{AEI} = \angle \text{AGI} = \text{90}^{\circ}\) angles
Hence \(\triangle \text{AEI} \cong \triangle \text{AGI}\)
So, by CPCT side \(\text{AE} = \text{AG}\)
Similarly, \(\text{CG} = \text{CF}\) and \(\text{BF} = \text{BE}\).
Property 2: If I is the incenter of the triangle, then \(\angle \text{BAI} = \angle \text{CAI}\), \(\angle \text{ABI} = \angle \text{CBI}\), and \(\angle \text{BCI} = \angle \text{ACI}\).
Proof: The triangles \(\text{AEI}\) and \(\text{AGI}\) are congruent triangles by RHS rule of congruency.
We have already proved these two triangles congruent in the above proof.
So, by CPCT \(\angle \text{BAI} = \angle \text{CAI}\).
Property 3: The sides of the triangle are tangents to the circle, hence \(\text{OE = OF = OG} = r\) are called the inradii of the circle.
Property 4: If \(s = \dfrac{a + b + c}{2}\), where \(s\) is the semiperimeter of the triangle and \(r\) is the inradius of the triangle, then the area of the triangle is: A = sr.
Property 5: Unlike an orthocenter, a triangle's incenter always lies inside the triangle.
Incenter Formula
To calculate the incenter of a triangle with 3 cordinates, we can use the incenter formula. Let us learn about the formula. Consider the coordinates of incenter of the triangle ABC with coordinates of the vertices, \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\) and sides \(a, b, c\) are:
\[(\dfrac{ax_1 + bx_2 + cx_3}{a + b + c}, \dfrac{ay_1 + by_2 + cy_3}{a + b + c})\]
Incenter of a Triangle Angle Formula
To calculate the incenter of an angle of a triangle we can use the formula mentioned as follows:
Let E, F, and G be the points where the angle bisectors of C, A, and B cross the sides AB, AC, and BC, respectively.
Using the angle sum property of a triangle, we can calculate the incenter of a triangle angle.
In the above figure,
∠AIB = 180° – (∠A + ∠B)/2
Where I is the incenter of the given 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. Here are the steps to construct the incenter of a triangle:
 Step 1: Place one of the compass's ends at one of the triangle's vertex. The other side of the compass is on one side of the triangle.
 Step 2: Draw two arcs on two sides of the triangle using the compass.
 Step 3: By using the same width as before, draw two arcs inside the triangle so that they cross each other from the point where each arc crosses the side.
 Step 4: Draw a line from the vertex of the triangle to where the two arcs inside the triangle cross.
 Step 5: Repeat the same process from the other vertex of the triangle.
 Step 6: The point at which the two lines meet or intersect is the incenter of a triangle.
Examples on Incenter

Example 1: If \(I\) is the incenter of the triangle \(\text{ABC}\) then find the value of \(x\) in the figure.
Solution:
Given:
I is the incenter of the triangle.
AI, BI, CI are the angle bisectors of the triangle, hence:
\[\begin{align}\angle \text{BAI} + \angle \text{CBI} + \angle \text{ACI} &= \frac{180^{\circ}}{2}\\[0.2cm]
37^{\circ} + 20^{\circ} + x^{\circ} &= 90^{\circ}\\[0.2cm]
57^{\circ} + x^{\circ} &= 90^{\circ}\\[0.2cm]
x^{\circ} &= 90^{\circ}  57^{\circ}\\[0.2cm]
x^{\circ} &= 33^{\circ}\end{align}\]Therefore, x = 33°.

Example 2: Peter calculated the area of a triangular sheet as 90 feet^{2}. The perimeter of the sheet is 30 feet. If a circle is drawn inside the triangle such that it is touching every side of the triangle, help Peter calculate the inradius of the triangle.
Solution:
Given:
The area of the sheet = 90 feet^{2}
The perimeter of the sheet = 30 feet
Semiperimeter of the triangular sheet =30 feet/2 = 15 feet
The area of the triangle = sr, where r is the inradius of the triangle.
Area = sr
90 = 15 × r
r = 90/15
r = 6Therefore, r = 6 feet.

Example 3: The coordinates of the incenter of the triangle ABC formed by the points A(3, 1), B(0, 3), C(3, 1) is (p, q). Find (p, q).
Solution:
Given:
The vertices of the triangles are A(3, 1), B(0, 3), C(3, 1).
\(\text{c} = \text{AB} = \sqrt{{(3  0)}^{2} + {(1  3)}^{2}}\)
\(\text{c} = \text{AB} = \sqrt{{3}^{2} + {2}^{2}} = \sqrt{\text{13}}\)
\(\text{a} = \text{BC} = \sqrt{{(3  0)}^{2} + {(1  3)}^{2}}\)
\(\text{a} = \text{BC} = \sqrt{{3}^{2} + {2}^{2}} = \sqrt{\text{13}}\)
\(\text{b} = \text{AC} = \sqrt{{(3  3)}^{2} + {(1  1)}^{2}}\)
\(\text{b} = \text{AC} = \sqrt{{6}^{2} + {0}^{2}} = \text{6}\)
Incenter of the triangle is:
\[(\dfrac{ax_1 + bx_2 + cx_3}{a + b + c}, \dfrac{ay_1 + by_2 + cy_3}{a + b + c})\]
\[ (\dfrac{3 \sqrt{13} + 0  3 \sqrt{13}}{6 + 2 \sqrt{13}}, \dfrac{2 \sqrt{13} + 18}{6 + 2 \sqrt{13}})\]
\[ (0, \dfrac{2 \sqrt{13} + 18}{6 + 2 \sqrt{13}}) \]Therefore, \[\text{Coordinates} = (0, \dfrac{2\sqrt{13} + 18}{6 + 2\sqrt{13}})\].
FAQs on Incenter
What is the Incenter of a Triangle?
The incenter of a triangle is the point of intersection of all the three interior angle bisectors of the triangle. This point is equidistant from 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 a triangle's circle since the largest circle could fit inside a triangle. The circle that is inscribed in a triangle is called an incircle of a triangle.
How to Find the Incenter of a Triangle?
For a triangle, an incenter can be obtained by drawing the angle bisectors of the triangle and locate the point of intersection of these bisectors. This can be done by using a compass. The steps to draw the angle bisectors are mentioned below:
 Step 1: Place one of the compass's ends at one of the triangle's vertex. The other side of the compass is on one side of the triangle.
 Step 2: Draw two arcs on two sides of the triangle using the compass.
 Step 3: By using the same width as before, draw two arcs inside the triangle so that they cross each other from the point where each arc crosses the side.
 Step 4: Draw a line from the vertex of the triangle to where the two arcs inside the triangle cross.
 Step 5: Repeat the same process from the other vertex of the triangle.
 Step 6: The point at which the two lines meet or intersect is the incenter of a triangle.
What Does Incenter Mean?
Incenter is the point where three bisectors of the interior angles of a triangle intersect and it is the center of the inscribed circle.
Is the Incenter Always Inside the Triangle?
Yes, the incenter is always inside the triangle. It is the point forming the origin of a circle that is inscribed inside the triangle. Just like a centroid, an incenter is always inside the triangle and it is made by taking the intersection of the angle bisectors of all three vertices of the triangle.
What is the Difference Between Orthocenter and Incenter?
An incenter is a point where three angle bisectors from three vertices of the triangle meet. That point is also considered as the origin of the circle that is inscribed inside that circle. Whereas an orthocenter is a point where three altitudes of the triangle intersect.
What is the Difference Between Centroid, Orthocenter, Circumcenter, and Incenter?
A circumcenter is a point that is equidistant from all the vertices of the triangle and it is denoted as O. An incenter is the point that is equidistant from the sides of the triangle and it is denoted as I. An orthocenter is a point where all the altitudes of the triangle intersect and it is denoted as H. A centroid is the point of inspection of the medians of the triangles and it is denoted by G.
What is Circle Incenter?
A circle incenter is the center of the triangles circle that is inscribed inside the triangle. The largest circle inscribed in a triangle will fit the triangle accurately by touching all three sides of the triangle.
What is the Incenter of a Triangle Angle Formula?
Let E, F, and G be the points where the angle bisectors of C, A, and B cross the sides AB, AC, and BC, respectively. The formula is ∠AIB = 180° – (∠A + ∠B)/2.