How to find the incenter of a triangle? – Step by step

The incenter of a triangle can be found by sketching the angle bisectors of the triangle and finding their point of intersection. In addition, we can also calculate the coordinates of the incenter using a formula with the coordinates of the vertices and the lengths of the sides of the triangle.

In this article, we will learn how to find the incenter of a triangle using a graphical method and an algebraic method. Then, we will solve some practice problems.

GEOMETRY
find the incenter of a triangle graphically

Relevant for

Learning to find the incenter of a triangle.

See methods

GEOMETRY
find the incenter of a triangle graphically

Relevant for

Learning to find the incenter of a triangle.

See methods

What is the incenter of a triangle?

The incenter of a triangle is the point of intersection of the three bisectors of the triangle. In turn, the bisectors are the segments that divide the angles in half as we can see in the diagram:

incenter of a triangle
incenter of a triangle with an incircle

Alternatively, we can define the incenter of a triangle as the center of a circle inscribed in the triangle. In turn, an inscribed circle is the largest circle that fits inside the triangle.


Finding the incenter of a triangle graphically

We can find the incenter of a triangle graphically by drawing the angle bisectors and finding the point of intersection. Therefore, we follow the following steps using a compass:

Step 1: Center the compass at vertex B and using any radius, draw an arc that cuts both sides of the triangle. Therefore, we get points D and E.

Step 2: With the same radius, center the compass at points D and E to draw two arcs to get the point of intersection F.

Step 3: Draw a segment that passes through points B and F. That segment is the bisector of angle B.

Step 4: By repeating the same process using another vertex of the triangle, we can draw another bisector.

Step 5: Find the point of intersection of two bisectors.

The point of intersection represents the incenter of the triangle.

find the incenter of a triangle graphically

Finding the incenter of a triangle algebraically

The coordinates of the incenter of the triangle can be obtained algebraically using a formula. However, we need to know the coordinates of all three vertices and the lengths of all three sides.

Suppose we have the following triangle:

incenter of a triangle with vertices

Then, we can calculate the incenter using the following formula:

$$\left(\frac{ax_{1}+bx_{2}+cx_{3}}{a+b+c},~\frac{ay_{1}+by_{2}+cy_{3}}{a+b+c}\right)$$

where, $latex A(x_{1},~y_{1})$, $latex B(x_{2},~y_{2})$ and $latex C(x_{3},~y_{3})$ are the coordinates of the three vertices of the triangle and a, b, c are the sides opposite each vertex.


Incenter of a triangle – Examples with answers

In the following examples, we can see how to find the coordinates of the incenter of a triangle algebraically.

EXAMPLE 1

I represents the incenter of the following triangle. What is the size of angle x?

example 1 of incenter of a triangle

Solution: Since I is the incenter of the triangle, the segments connecting the incenter to the vertices are bisectors, so they divide the angles into two equal parts.

This means that the three interior angles of the triangle are:

2×35°=70°

2×31°=62°

x=2x

A triangle always has a sum of interior angles equal to 180°, so we have:

70°+62°+2x=180°

2x=180°-70°-62°

2x=48°

x=24°

EXAMPLE 2

A triangle has an area of 15 m² and its perimeter is equal to 18 m. What is the radius of the circle inscribed in the triangle?

Solution: We have the following information:

  • Triangle area = 15 u²
  • Triangle perimeter = 18 u

The area of a triangle can be calculated using A=sr, where s is the semi perimeter and r is the inradius. The semi perimeter is half the perimeter. Therefore, we have:

A=sr

15=9r

r=1.67

Thus, the radius of the inscribed circle is 1.67 m.

EXAMPLE 3

The triangle below has the vertices A(0, 4), B(0, 1), and C(4, 4). What are the coordinates of its incenter?

example 3 of incenter of a triangle

Solution: We can use the incenter formula along with the coordinates of the vertices and the lengths of the sides.

Using the diagram, we can see that the length of b is 4 units and the length of c is 3 units. Therefore, we use the Pythagorean theorem with those two sides to find the length of a:

a²=b²+c²

a²=4²+3²

a²=16+9

a²=25

a=5

Using the incenter formula, we have:

$$\left(\frac{ax_{1}+bx_{2}+cx_{3}}{a+b+c},~\frac{ay_{1}+by_{2}+cy_{3}}{a+b+c}\right)$$

$$=\left(\frac{0+0+3(4)}{5+4+3},~\frac{5(4)+4(1)+3(4)}{5+4+3}\right)$$

$latex =(\frac{12}{12},~\frac{36}{12})$

$latex =(1,~3)$

The coordinates of the incenter are (1, 3).


Incenter of a triangle – Practice problems

Solve the following practice problems using the algebraic method to find the coordinates of the incenter.

A triangle has an area of 90 square units and a perimeter of 90 units. What is the radius of the circle inscribed in the triangle?

Choose an answer






Find the coordinates of the incenter of a triangle that has vertices A(4, 5), B(0, 1), C(7, 2), and sides a=7.07, b=4.24, and c=5.66.

Choose an answer







See also

Interested in learning more about the incenter, orthocenter, centroid, and circumcenter of a triangle? Take a look at these pages:

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