Assignment 4

Centers of Triangles

In this assignment, I explored the relationships between the centers of a triangle ABC and its medial triangle XYZ. First let's define the words that are pertinent to understanding this exploration.

CENTROID- The centroid (G) of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.

ORTHOCENTER- The orthocenter (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side.

CIRCUMCENTER- The circumcenter (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, C is on the perpendicular bisector of each side of the triangle.

MEDIAL TRIANGLE- The medial triangle is the triangle that is formed when you connect the midpoints of each side of the original triangle.

Now with these five definitions let's begin our exploration of the relationships between the centers of a triangle and its medial triangle. Let's begin by looking at three different cases: right (left), acute (middle), and obtuse (right).

It appears in each case that the medial triangle divides the original triangle into four triangles all of which are similar to the larger triangle and all of which are congruent to each other. This is indeed true and this implies that each smaller triangle is one fourth the area of the larger triangle. For a proof click here. From this we can also deduce that XY is parallel to BC, XZ is parallel to AC, and ZY is parallel to AB.

Some questions that come to mind that I will investigate:

What is the relationship between the centroid of the medial triangle and the centroid of the original triangle?

Now let's add the lines for the medial triangle.

It looks like the each of the segments connecting the the vertex of the angles to the midpoint of the opposite side of the original triangle go through the midpoints of the medial triangles. If this were true then the centroid would be the same for the medial triangle and the original triangle. Let's construct the midpoints using GSP and see what happens.

It looks like each midpoint does lie on the segment. Let's take the specific case of segment AZ and midpoint H. Why is it true that H lies on AZ? We know that XY is parallel to BC and furthermore we know that Z is the midpoint of BC (and H is of course the midpoint of XY), and finally both segment XY and segment BC subtend angle BAC. From these facts, we can conclude that AZ must intersect XY at H the midpoint of XY. Therefore segment HZ, GY, and IX are subsets of AZ, BY, and CX so there intersection point must be the same. In other words the centroid for both the medial triangle and the original triangle are the same point.

What is the relationship between the orthocenter and the circumcenter for a triangle and its medial triangle? Let's take a look.

I have constructed the circumcenter of the triangle along with the triangles medial triangle. Just from looking at the diagram it appears that the orthocenter of the medial triangle will correspond to the circumcenter of the original triangle. How do we know this? We have established that all sides of the medial triangle are parallel to a side of the original triangle. Also, we have constructed the perpendicular bisectors of the original triangle so they are perpendicular to each side of the medial triangle and since they are the bisectors of the original triangle they go through the vertices of the medial triangle. This means the perpendicular bisectors of the original triangle are the same as the altitudes of the medial triangle. (No graphical representation is given below as it would be the same picture as the one above. However, it would be helpful to have students draw the perpendicular bisectors of the original triangle and then draw the altitudes of the medial triangle and discover that they get no new lines and ask them to justify why this happened.)

Now let's look at a graph with the circumcenter of the medial triangle and the orthocenter of the original triangle, the centroid of both triangles as well as this shared point from above.

As you can see the orthocenter of the medial triangle and the orthocenter of the original triangle lie on opposite sides of the centroid and the same thing is true of the circumcenters. This happens because of the orientation of the medial triangle which is similar to the original triangle but is rotated 180 degrees in two different directions. Also we can conclude that the Euler line (the line which is incident with the circumcenter, orthocenter, and centroid of a given triangle) is the same for the medial triangle as it is for the original triangle we know this because the two triangles share some of the same centers which means that there Euler line has to be the same. This concludes the study of the centers of medial triangles of a given triangle.