Today we’re going to be exploring the different centers of a triangle. To show this we’re going to construct different kinds of triangles using Geogebra! The four centers that we will be discussing today are the circumcenter, incenter, centroid, and orthocenter. By exploring our different constructions for these different triangle constructions, we’ll be forming our own conjectures about where these different centers lie in the triangle (the answer may not be so obvious as the name implies). Finally, we’ll be putting all of our centers together in one circle so that we can examine their position relative to one another.
Before we start exploring the centers of our triangles, let’s review what each of our triangles look like!
Let’s first look over the scalene triangle. A scalene triangle is simply a triangle with three unequal sides. Below is an example construction of this type of triangle using Geogebra (Figure 1). Notice how the length of each side of the triangle is different from one another, therefore making it a scalene triangle.
Next, let’s take a look at the isosceles triangle. An isosceles triangle is a triangle with at least two congruent sides which thereby create two congruent angles. In Figure 2 below notice how this is reflected in the measures of our angles and sides. If you are already not familiar with this notation, look at the black hash marks and angle arcs that denote the congruent angles and sides of the triangle.
The next triangle we’ll be looking at is the equilateral triangle. As the same suggests, the equilateral triangle has all sides and angles congruent to one another. This can be seen in Figure 3 below.
For acute, right, and obtuse angles, we look at the angle measure to determine how to categorize an angle. An angle with less than 90 degrees is an acute angle. An angle with a measure equal to 90 degrees is a right angle, and an angle with a measure more than 90 degrees is an obtuse angles. Examples of these types of angles are given in Figure 4.
Now that we’ve reviewed our knowledge of the different types of triangles we’ll be working with, let’s explore the different centers of a triangle.
For our first center, lets look at the circumcenter! If you’re not familiar with what a circumcenter is, it’s the point of a triangle where the perpendicular bisectors meet. The circumcenter is also the center of the triangle’s circumcircle. This circumcircle is a circle which passes through all of the triangle’s vertices To look at what this circumcenter will look like, let’s construct some different triangles and observe where the circumcenter lies either in or out of the triangle. The triangles we will be dealing with are scalene, isosceles, equilateral, acute, right, and obtuse. In the chart below, let’s add our conjectures for each type of triangle with respect to the position of the circumcenter.
| Scalene | Isosceles | Equilateral |
| The circumcenter will always remain inside the scalene triangle unless one of the angles of the triangle is over 90 degrees. | Similar to the scalene triangle, the circumcenter will remain in the isosceles triangle unless one of the angles of the triangle is over 90 degrees. | The circumcenter will remain in the center of the triangle, equidistant from all three vertices. |
| Acute | Right | Obtuse |
| The circumcenter will always remain inside any triangle as long as the angle is acute. | At 90 degrees, the circumcenter lies on the midpoint of the hypotenuse. | For an obtuse angle, the circumcenter will go outside of the triangle. |
To better see some of these conjectures, visit this link to an online Geogebra sketch.
Now that we’ve looked at the circumcenter of the triangle, let’s move on to our next center, the incenter. The incenter of the triangle is the point where the angle bisectors all intersect. Again, similiar to the circumcenter, the incircle of the triangle is also the center of the incircle, the largest circle that will fit inside the triangle and touch all three sides. An important property to note for the incircle is that it will never go outside of the triangle. Like with the circumcenter, let’s construct some different triangles to create some conjectures about the position of the incenter in a triangle. I encourage you to follow along and explore these conjectures yourself by using this link to a different online Geogebra sketch.
| Scalene | Isosceles | Equilateral |
| The position of the incenter will be inside of the triangle. | The position of the incenter will be inside of the triangle. The angle bisector that goes through the non-base angle goes through the midpoint of the base. | Angle bisectors go through all midpoints of the three sides. The incenter is equidistant from the three vertices of the triangle. |
| Acute | Right | Obtuse |
| The position of the incenter will be within in the triangle. | The position of the incenter will be within the triangle. | The position of the incenter will be within the triangle. |
The third center that we will explore is the centroid of the triangle. The centroid of the triangle is the point where the three medians of the triangle meet. The median of the triangle is the segment joining a vertex to the midpoint of the side opposite to it. The centroid is also called the “center of mass” since it is the point where all of the mass for triangle acts. In addition, the centroid is always within the triangle. Like with the circumcenter and incenter, let’s construct some different triangles to create some conjectures about the position of the centroid in a triangle. I encourage you to follow along and explore these conjectures yourself by using this link to a different online Geogebra sketch.
| Scalene | Isosceles | Equilateral |
| Centroid is always within the triangle. | Line through the median of the base and the vertex of the non-base angle creates two pairs of congruent segments from the lines through the medians. | Occurs at the same point as the other centers. |
| Acute | Right | Obtuse |
| Stays within the triangle. | Stays within the triangle. | Stays within the triangle. |
The fourth and final center that we will explore is the orthocenter of the triangle. The orthocenter is the point where the altitudes of the triangles meet. The altitude of a triangle is a segment that goes through a vertex and is perpendicular to the opposite side of the vertex. In addition, the orthocenter is not always inside of the triangle. Like with the previous centers, let’s construct some different triangles to create some conjectures about the position of the orthocenter in a triangle. I encourage you to follow along and explore these conjectures yourself by using this link to a different online Geogebra sketch.
| Scalene | Isosceles | Equilateral |
| The position of the orthocenter is inside of the triangle for acute angles. | The altitude coming from the non-base vertex splits the other two altitudes into two congruent segments. | Occurs at same point as other centers. |
| Acute | Right | Obtuse |
| Orthocenter is within the triangle . | Orthocenter is at the vertex of the 90 degree angle. | Orthocenter is outside of the triangle. |
Now that we’ve looked at the different centers in our different triangles, let’s put it all together and see how the different centers relate to one another in our triangles. Similar to before, look at this link to see an online Geogebra sketch with the different centers labeled.
| Scalene | Isosceles | Equilateral |
| All of the centers will stay inside of the scalene triangle if the angles are acute. | All of the centers will stay in the middle of the isosceles triangle if the angles are acute. | All of the centers will be equidistant from the three vertices of the equilateral triangle. |
| Acute | Right | Obtuse |
| If the angle is acute, the centers of triangle will stay inside of the triangle. | If there is a right angle in the triangle, the orthocenter will be the vertex of the 90 degree angle. The circumcenter will be the midpoint of the hypotenuse and the centroid and incenter will stay inside of the triangle. | If there is an obtuse angle, the circumcenter and the orthocenter will be outside of the triangle. The centroid and the incenter will stay inside of the triangle. |
From this, we can conclude that if there is a right or obtuse any in any of our triangles, then the circumcenter and the orthocenter will be either outside of the triangle or on top of a vertex or midpoint. Further, the centroid and incenter will always be inside of the triangle, regardless of what angles compose the triangle.
Follow this link to look at the teacher discussion for this project!
Morrell Math Corner 