## Explorations of Triangle Centers

### Marianne Parsons

Using Geometer's Sketchpad 4.06 we can explore the different centers of any given triangle. Regardless of the type of scalene triangle, as described below, the relationship between these centers remains constant.

In this exploration we will be examining three different types of scalene triangles- obtuse, acute, and right. In each type of triangle, the sum of angles from each vertices is 180 degrees. It is the composition of those angles that determine whether the triangle will be classified as obtuse, acute, or right.

An obtuse triangle is defined as a triangle where one of the vertices has an angle that is GREATER THAN 90 degrees.

In this image, we can see the angle at vertex B is greater than 90 degrees.

A right triangle is defined as a triangle where one of the vertices has an angle that is EQUAL to 90 degrees.

In this image we can see the angle at vertex B is 90 degrees.

An acute triangle is defined as a triangle where each of the vertices has an angle that is LESS THAN 90 degrees.

In this image, we can see all of the angles in the triangle are less than 90 degrees.

### Centroid

First, let's examine the centroid, point P, of our triangles. The centroid is defined by the intersection of the segments drawn from each vertex to the midpoint of the opposite side. These segments are called the medians of our triangles ABC. The centroid is the intersection of these medians.

### Orthocenter

Next, let's examine the orthocenter, point O, of our triangles. The orthocenter is defined as the intersection of the three altitudes of our triangles ABC. The altitudes are drawn from the vertices of triangle ABC to a point that is perpendicular to the opposite side.

### Circumcenter

Now, let's examine the circumcenter, point S, of our triangles. The circumcenter is defined by the intersection of the perpendicular bisectors of our triangles ABC. The perpendicular bisectors are drawn as perpendicular lines going through the midpoints of each side.

### Incenter

Finally, let's examine the incenter, point I, of our triangles. The incenter is defined by the intersection of the angle bisectors of our triangles ABC. The angle bisectors are drawn from the vertices of our triangle, and bisect the angles into two equal parts.

Click here to see an animation of our four centers of a triangle, as the triangle moves from obtuse, to right, to acute.

To view the animation you will need to own Geometer's Sketchpad 4.06.

### What can we observe?

The centroid P always lies on the inside of the triangle. This is something we expect because the centroid is defined by the intersection of the triangles medians.

The incenter I always lies on the inside of the triangle as well. This is also something we can expect since the definition of incenter is the intersection of the angle bisectors of each vertices.

Regardless of the shape of our triangle, our circumcenter S, centroid P, and orthocenter O, will always form a straight line. This line is called Euler's Line. The distance between the centers will vary as we adjust the shape of our triangle, however they will always remain in a straight line.