Kristen Robinson

EMAT 4680

7 September 2000

Homework 8 Problem 7

An orthic triangle is formed using the feet of the altitudes of a larger triangle as the vertices of the orthic triangle. Constructing the four centers of the triangle provides an opportunity for comparison between the centers of the orthic triangle and the centers of the original triangle. No real similarities exist between the two sets of centers beyond the basic descriptions and patterns of the four individual centers. However, some comparisons could be made.

The 2nd incenter lies on the same point as the orthocenter. The orthocenter is the intersection of the altitudes of the original triangle. The 2nd incenter is the intersection of the angle bisectors of the orthic triangle. For example, the altitude from A to BC is point E, and the angle bisector of angle DEF is the segment from E to A. These two segments are the same. Therefore, the intersection points which construct these two centers are the same intersection point.

Figure 1.

The second comparison to be made is that the 2nd circumcenter moves in a pattern very similar to the incenter, and it lies on a point very close to that of the incenter. Often, these two centers lie on the same point.

Figure 2.

Both the 2nd centroid and the 2nd orthocenter follow the basic guidelines for centroids and orthocenters, but neither has a direct relationship with any center from the original triangle.

Figure 3.

Furthermore, the restriction that the original triangle be an acute triangle because the altitudes of the original triangle do not exist if the original triangle is either right or obtuse.

Figure 4.

Figure 5.