The incenter of a triangle is a point that represents the intersection of the three bisectors of a triangle. Furthermore, the incenter can also be considered as the center of a circle inscribed in the triangle. We can find the coordinates of the incenter using a formula.

In this article, we will learn about the incenter of a triangle in detail. We will learn about its formula and we will use it to solve some practice problems.

## Definition of the incenter of a triangle

The incenter of a triangle is the point where the bisectors of the three interior angles of the triangle intersect. In turn, let us remember that the bisectors are the lines or segments that divide an angle into two equal parts. The following is a diagram of the incenter of a triangle:

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

The incenter is always located inside the triangle no matter what type of triangle we have. An important feature of equilateral triangles, however, is that the incenter, orthocenter, circumcenter, and centroid are in the same position.

## Properties of the incenter of a triangle

**Property 1:** The incenter of a triangle is always located inside the triangle no matter what type of triangle we have.

** Property 2:** If

*I*is the incenter of the triangle, then segments AE and AF must have the same length. The same happens with the segments BE and BG, and with the segments CG and CF.

** Property 3:** If

*I*is the incenter of the triangle, then the angles ∠ABI and ∠CBI are equal. The same happens with the angles ∠BAI and ∠CAI, and with the angles ∠ACI and ∠BCI.

** Property 4:** The sides of the triangle are tangent to the inscribed circle, so IE, IF and IG are equal to the radius of the circle and are called the inradius.

** Property 5:** The area of the triangle can be calculated using the formula

*A=sr*, where

*r*is the inradius of the triangle and

*s*is the semi perimeter. In turn, the semi perimeter is $latex s=\frac{a+b+c}{2}$.

## Formula for the incenter of a triangle

We can calculate the coordinates of the incenter of a triangle using the incenter formula. To illustrate this formula we will use the following triangle ABC.

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

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.

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## Finding the incenter of a triangle graphically

If we don’t know the coordinates of the vertices of the triangle, we can find the incenter graphically. We can accomplish this by sketching the bisectors and finding the point of intersection.

We can follow the following steps using a compass:

**Step 1:** Use vertex B as the center and any radius to draw an arc and cut both sides of the triangle. In this way, we will get points D and E.

** Step 2:** Use the same radius from points D and E and draw two arcs to get the point of intersection F.

** Step 3:** Draw a line from point B to point F. That line is the bisector of angle B.

** Step 4:** Repeat the same process using another vertex of the triangle to draw another bisector.

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

The two lines drawn are two bisectors of the triangle. Therefore, the point of intersection is the incenter of the triangle.

## Incenter of a triangle – Examples with answers

The following examples are solved by applying what we have learned about the incenter of a triangle.

### EXAMPLE 1

Determine the size of angle *x* if *I* represents the incenter of the following triangle.

**Solution:** Since *I* is the incenter of the triangle, we know that the segments that divide the angles are the bisectors. Therefore, the segments divide the angles into two equal parts.

This means that the three angles of the triangle are:

2×35°=70°

2×31°=62°

2×*x*=2*x*

The interior angles of a triangle always add up to 180°, so we have:

70°+62°+2*x*=180°

2*x*=180°-70°-62°

2*x*=48°

*x*=24°

### EXAMPLE 2

We have that the area of a triangle is 15 square units and its perimeter is equal to 18 units. If we draw an inscribed circle, what is the radius of the circle?

**Solution:** We have the following information:

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

We know that the area of a triangle can be calculated as *A=sr*, where *s* is the semiperimeter and *r* is the inradius. From the given information, we can easily calculate the semiperimeter, which is 9 units. Therefore, we have:

*A=sr*

15=9*r*

*r*=1.67

Therefore, the radius of the inscribed circle is 1.67 units.

### EXAMPLE 3

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

**Solution:** To calculate the coordinates of the incenter, we need the coordinates of the vertices and the lengths of the sides.

From the diagram, we can easily deduce that the length of *b* is 4 units and the length of *c* is 3 units. Using these lengths and the Pythagorean theorem, we can find the length of *a*:

*a*²=*b*²+*c*²

*a*²=4²+3²

*a*²=16+9

*a*²=25

*a*=5

Now, we can use the incenter formula:

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

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

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

$latex =(1,~3)$

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

## See also

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

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