Assignment 4: Centers of a Triangle
In this writeup, I will present four different centers of a triangle, state some of the relationships between those centers, and construct the nine point circle.
Assignment 4 begins with directions on how to construct several centers. The centroid was the first center we expored. The centroid is pt. G in the diagrams below. The orthocenter is labeled pt. H. Point C is the circumcenter. The circumcircle of triangle ABD can be consturcted by using point C as the center and any vertex as a point of the circle. The incenter is point I. Once we find the incenter, we can also find the incircle.
Now, we can compare the four centers in one triangle.
We were asked to look for relationships among these centers. It was revealed that G,H, and C are colinear. It is hard to see this relationship from triangle ABD. I changed the triangle and looked at the centers.
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Our next task was to construct three types of triangles, the medial triangle, the orthic triangle and the orthocenter, mid-segment triangle.
I wanted to compare G, H, C and I of each on the constructed triangles with the original triangles. The pink points represent the centers of the original triangle and are lableed with letters. The green points represent the centers of the consturcted triangles and they are labled as prime letters. As you can see, sometimes the two triangles share a common center. We were given some facts about these constructed triangles. The medial triangle is similar to the orginal triangle and it is one-fourth of its area.The orthocenter mid-segment triangle is congruent to the medial triangle and similar to the original triangle. In the medial triangle, the centroid of the original triangle is the same point as the centroid of the constructed triangle. Also in the medial triangle the circumcenter of the original triangle is equal to the orthocenter of the consturcted triangle. In the orthocenter, mid-segment triangle, the orthocenter is the same for both the original triangle and the consturcted triangle. The orthocenter of the original triangle is the same point as the incenter of the constructed triangle for the ortic triangle.
Our next task was to construct the nine point circle. We were given that the nine-point circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments for the respective vertices to orthocenter. So I labled a triangle to be my given triangle. Then I found the nine points mentioned above. I knew I had to find the center of the circle inorder to construct the circle. After several explorations on other problems, I discovered that all I needed to do to find the nine point circle was to circumstribe the medial triangle. So, I constructed the medial triangle and found the circumcenter of that triangle. The I consturcted the circle and that constructed circle did in deed pass through all nine of the points mentioned above.
