The **CENTROID**, **G**, of a triangle
is the common point of intersection of the three medians of a
triangle. Recall, that a **median** of a triangle is the segment
from a vertex to the midpoint of the opposite side. It is important
to note the centroid is actually the *center of gravity*
of the triangle. This means that if you make a "real"
triangle out of cardboard, you can balance the triangle at this
point.

To construct the centroid of a triangle, first construct a triangle ABC. Then construct the midpoints of three segments.

Next, construct a line segment from the vertices of the triangle to the midpoints of the opposite sides.

The intersection point of all three medians
is the **centroid, G** of triangle ABC.

Is the centroid always located inside of the triangle?

Use the script in GSP, by clicking on __Centroid
Script__ to construct different triangles to explore the location
of G.

Note: You must have Geometers Sketch Pad to run the script.

**Centroid Script:** Obtain a new sketch in GSP. Input
points A, B, C. Then click on **play** on the script. Once
the script has constructed the centroid, drag one point around
to change the size of the triangle. Notice the location of G.

Notice: No matter the shape--(obtuse, acute, or right) or size of the triangle, the centroid always stays inside, and in the "center" of the triangle.

The **ORTHOCENTER, H** of
a triangle is the common point of intersection of the three lines
containing the altitudes. Recall that an altitude is a perpendicular
segment from a vertex to the line of the opposite side.

To construct the orthocenter of a triangle. Construct triangle ABC. Then construct the altitudes from two vertices to the opposite sides by first constructing lines AB, BC, and AC. Be sure to construct the perpendicular lines from the vertices to the opposite sides (extended) and not the segments.

Notice that the orthocenter may not be located inside of triangle ABC.

Is the orthocenter ever inside of the triangle?

Use the script in GSP, by clicking on __Orthocenter
Script__ below to construct different triangles to explore the
location of H.

Note: You must have Geometers Sketch Pad to run the script.

**Orthocenter Script:** Obtain a new sketch in GSP. Input
points A, B, C. Then click on **play** on the script. Once
the script has constructed the centroid, drag one vertex of the
triangle around to change the size of the triangle. Notice the
location of H.

Notice: If ABC is obtuse, H goes outside of the triangle, if ABC is acute, H stays inside, and if ABC is right, H is the vertex opposite of the hypotenuse of the triangle.

The **CIRCUMCENTER, C**
of a triangle is the common point of intersection of the perpendicular
bisectors of each side of a triangle. It is the point in the plane,
which is equidistant from the three vertices of the triangle.
Recall that a perpendicular bisector is perpendicular to a side
of the triangle and goes through the midpoint of that side.

To construct the circumcenter of a triangle. Construct triangle ABC. Then construct the midpoints of two sides of the triangle. Next, construct lines perpendicular to the sides and through the midpoints of the sides.

Notice that in this case, the circumcenter is located inside triangle ABC.

Is the orthocenter ever outside of the triangle?

Use the script in GSP, by clicking on __Circumcenter
Script__ below to construct different triangles to explore the
location of C.

Note: You must have Geometers Sketch Pad to run the script.

**Circumcenter
Script:** Obtain
a new sketch in GSP. Input points A, B, C. Then click on **play**
on the script. Once the script has constructed the centroid, drag
one vertex of the triangle around to change the size of the triangle.
Notice the location of C.

Notice: If ABC is obtuse, C goes outside of the triangle, if ABC is acute, C stays inside, and if ABC is right, C is the midpoint of the hypotenuse of the triangle. Notice, also that C exits the triangle through the mid-point of a side of the triangle.

Now let's explore the relationship among all three of the centers discussed above. Below is a triangle with all three centers.

Notice that all three of the centers are collinear. Does this always happen or is this a special case?

For a convincing argument that they are collinear,
use the script in GSP, by clicking on __Euler Script__ below
to construct different triangles and explore the relationships
of the centroid, the orthocenter, and the circumcenter.

Note: You must have Geometers Sketch Pad to run the script.

**Euler Script:** Obtain a new sketch in GSP. Input
points A, B, C. Then click on **play** on the script. Once
the script has constructed the centers, drag one vertex of the
triangle around to change the size of the triangle. Notice the
location of all three centers. Observe that the three centers
remain collinear, reguardless of the size or shape of the triangle.
Notice, also that as the circumcenter exits throught the midpoint,
the orthocenter exits through the opposite vertex, similtaneously
(See picture below). Recall: the centroid remains inside of the
triangle.