Assignment 4: Problem 6

 

Jason P. Pickhardt


Problem:

 

Take any triangle. Construct a triangle connecting the three midpoints of the sides. This is called the MEDIAL triangle. It is similar to the original triangle and one-fourth of its area. Construct G, H, C, and I for this new triangle. Compare to G, H, C, and I in the original triangle.

 

Exploration:

 

This exploration is best suited for the use of Geometerís Sketchpad. Thus all of the work was done in GSP here.

 

Conclusion:

 

As can be seen in the GSP sketch there are several observations to make regarding the relationship between the centers of a triangle and the center of its medial triangle. Some observations/conclusions that can be drawn are:

 

1)      The centroid, orthocenter and circumcenter of both triangles are collinear.

2)      The incenter and centroid of both triangles are collinear.

3)      When all eight centers are collinear both the original triangle and the medial triangle are isosceles.

 

Jasonís Homepage