Write Up # 4

Centers of a Triangle

by: Angel Rowe Abney

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.