Incenter of a Triangle is the intersection point of all the three angle bisectors of a Triangle. The incenter is an important point in a triangle where lines cutting angles in half come together. This point is also the center of a circle called Incircle that fits perfectly inside the triangle and touches all three sides the same. This article covers various concepts of the incenter of the triangle, such as why this point is important, how to find it using a compass or numbers, and properties of the incenter of the circle.
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What is Incenter of a Triangle?
The incenter of a triangle, as the name suggests, is the center point of the triangle. This point which we call an incenter forms at the junction where all the lines that bisect the inner angles meet together. The distance of the point from all three sides of the triangle are same. The incircle of the triangle also fits a perfect circle inside the triangle and this circle is called the incircle of the triangle.
Incenter Definition
The incenter of a triangle is the point inside the triangle where all three lines that cut its inside angles in half come together. This point is the same distance from the three sides of the triangle, making it like the triangle's middle. It's also the center of the largest circle that can fit snugly inside the triangle, which we call the "incircle." To symbolize the incenter, we typically use the letter "I,"
Properties of an Incenter of a Triangle
Some important properties of the incenter of the triangle are given below:
Property 1: If I is the incenter of a triangle ABC, then three pairs of line segments are equal in length: AE and AG, CG and CF, and BF and BE. This means that AE = AG, CG = CF, and BF = BE.
Property 2: The incenter I also has a special relationship with the angles of the triangle. It causes the angles ∠BAI and ∠CAI to be equal, ∠BCI and ∠ACI to be equal, and ∠ABI and ∠CBI to be equal. This follows the angle bisector theorem.
Property 3: The incenter I is the center of a circle that touches all three sides of the triangle, and the distances from I to the sides of the triangle (EI, FI, GI) are all the same. These distances are called the inradii, or the radius of the incircle.
Property 4: You can calculate the area of the triangle using the semiperimeter (s) and the inradius (r). The formula is A = sr, where A is the area, s is the semiperimeter (s = (a + b + c)/2, where a, b, and c are the side lengths of the triangle), and r is the inradius.
Property 5: The incenter of a triangle always stays inside the triangle. Unlike the orthocenter, which can be outside the triangle in some cases, the incenter is always contained within the boundaries of the triangle.
Incenter of a Triangle Formula
The formula to find the incenter of the formula with 3 coordinates (x1, y1), (x2, y2), and (x3, y3) is:
{(ax1 + bx2 + cx3)/(a + b + c), (ay1 + by2 + cy3)/(a + b + c)}
In simple terms, to get the incenter, you:
- Multiply the x-coordinate of point A by side length a, the x-coordinate of point B by side length b, and the x-coordinate of point C by side length c. Then, add these together.
- Divide the result by the sum of side lengths a, b, and c.
- Repeat the same process for the y-coordinates, but using side lengths a, b, and c.
Incenter of a Triangle Angle Formula
The formula to find the incenter of an angle of a triangle is as follows:
Let, In a triangle D, F, and G are the points where the angle bisectors of angles A, B, and C respectively, meet the sides BC, AC, and AB.
The angle ∠AIB (where I is the incenter of the triangle) can be calculated using the formula:
∠AIB = 180° – (half of the sum of angles A and B)
OR
∠AIB = 180° – (∠A + ∠B)/2
How to Find Incenter of a Triangle
There are two methods for finding the incenter of a triangle. In construction, we locate the incenter by drawing the triangle's angle bisectors. In coordinate geometry, we employ a formula to determine the incenter.
Using Coordinate Geometry: Find the incenter of the triangle with the coordinates given as: A(2, 2), B(6, 2), and C(4, 5)
According to the given information
- (x1, y1) = (2, 2)
- (x2, y2) = (6, 2)
- (x3, y3) = (4, 5)
We know the incenter of a triangle is:
I(x, y) = {(ax1 + bx2 + cx3)/(a + b + c), (ay1 + by2 + cy3)/(a + b + c)}
For side a: The distance between points B and C = √((6 - 4)2 + (2 - 5)2) = √8
For side b: The distance between points A and C = √((2 - 4)2 + (2 - 5)2) = √13
For side c: The distance between points A and B = √((6 - 2)2 + (2 - 2)2) = 4
Putting the values of a, b, c in the formula of incenter, we get:
I(x, y) = {(8×2 + 13×5 + 4×4)/(8 + 13 +4), (8×2 + 13×2 + 4×5)/(8 + 13 +4)}
⇒ I(x, y) = (16 + 78 + 16)/(25), (16 + 26 + 20)/(25)
⇒ I(x, y) = (110/25, 62/25) = (22/5,62/25)
∴ The incenter of the triangle ABC with the coordinates is (22/5,62/25)
How to Construct the Incenter of a Triangle?
To construct the incenter of a triangle it will require to use a compass. Using a compass follow the below given steps:
Step 1: Put one end of the compass on a vertex of the triangle and the other end touches one side.
Step 2: Use the compass to draw two arcs on two sides of the triangle.
Step 3: With the same distance on the compass, make two arcs inside the triangle. These arcs should cross each other from where they touch the sides.
Step 4: Draw a line from the triangle's vertex to where the two inside arcs cross.
Step 5: Repeat the same steps from the other vertex of the triangle.
Step 6: Where the two lines meet or cross is the incenter of the triangle.
Incenter of Right-Angle Triangle
The incenter if a Right-angled triangle is the point where all the angle bisectors of a right angled triangle meet. If the sides of a right triangle measures a, b and c then radius of the incircle 'r' is given as r = (ab)/(a + b + c). The incenter of the right triangle is illustrated below:
Centroid, Circumcenter, Incenter, Orthocenter
Centroid, Circumcenter, Incenter and Orthocenter are the four important points related to a traingle. A comparison between Centroid, Circumcenter, Incenter and Orthocenter is tabulated below:
Centroid | Circumcenter | Incenter | Orthocenter |
|---|---|---|---|
Point of Intersection of Median | Point of Intersection of Perpendicular Bisector | Point of Intersection of Angle Bisector | Point of Intersection of Altitudes |
Divides Median in 2:1 | Center of Triangle's circumcircle | Center of Triangle's Incircle | Orthocenter of Right Triangle is at Right angle |
Also, Check
Examples of Incenter of a Triangle
Example 1: Calculate the Incenter of Triangle ABC. AB= 8cm, BC= 15 cm, CA= 17 cm.
Solution:
Using the formula of Incenter of Triangle = (aA + bB + cC)/(a + b + c)
where,
- a = 8
- b = 15
- c = 17
And Angles are,
- A = 30°
- B = 60°
- C = 90°
Putting these value in the formula to get,
⇒ {(8)(30) + (15)(60) + (17)(90)}/{8 + 15 + 17}
⇒ (240 + 900 + 1530)/40
⇒ 2670/40
⇒ 66.75
Example 2: Jane calculated the area of a triangular field as 120 square meters. The perimeter of the field is 36 meters. If a circle is drawn inside the triangle in a way that it touches every side of the triangle, help Jane calculate the inradius of the triangle.
Solution:
According to the given information,
Area of the triangle = 120 square meters
Perimeter of the triangle = 36 meters
We know, the Area of a Triangle = r × s
s = semiperimeter
s = p/2 = 36/2 = 18
A = r × s
r = A/s
r = 120/18
r = 6.67 meters
Practice Problems on Incenter of a Triangle
Problem 1: Given triangle PQR with vertices P(1, 2), Q(4, 6), and R(7, 2), find the coordinates of the incenter.
Problem 2: Construct a triangle ABC with ∠A = 45°, ∠B = 60°, and ∠C = 75°. Use the construction method to find the incenter.
Problem 3: In triangle LMN, if ∠L = 75°, ∠M = 60°, and ∠N = 45°, find the coordinates of the incenter.
Problem 4: Construct a triangle XYZ with ∠X = 80°, ∠Y = 50°, and ∠Z = 50°. Use the construction method to find the incenter.