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.

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.

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.

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.

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