Kristen Robinson

EMAT 4680

5 September 2000

Homework #7

Several relationships and patterns among the four centers of a triangle were discovered by observing the actions of the centers of various types of triangles. The three main types of triangles constructed depending upon the angles of the triangle are an acute, a right, and an obtuse triangle. With an acute triangle, all four centers lie inside the boundaries created by the three sides of the triangle.

With a right triangle, though, the orthocenter lies at the vertex of the right triangle. In addition, the circumcenter lies at the midpoint of the hypotenuse. The other two centers remain within the boundaries created by the triangle.

With an obtuse triangle, both the orthocenter and the circumcenter lie outside of the triangle. As usual, the incenter and the centroid still lie within the boundaries of the triangle.

Other observations can be made by observing the three types of triangles constructed depending upon the sides of the triangle. With a scalene triangle and acute angles, all four centers lie within the boundaries of the triangle (see Figure 1). With a scalene triangle and an obtuse angle, both the orthocenter and the circumcenter lie outside of the triangle(see Figure 3). Additionally, both the incenter and the centroid lie within the boundaries of the triangle. With a scalene triangle and a right angle, the orthocenter lies on the vertex of the right angle, and the circumcenter lies on the midpoint of the hypotenuse (see Figure 2). The remaining centers lie within the boundaries of the triangle.

With an isosceles triangle, all four centers lie within the boundaries of the triangle. This is because the angles of an isosceles triangle are all acute.

With an equilateral triangle, the most interesting relationship between the centers was noticed. All four centers lie at the same point within the triangle. This point is equidistant from the vertices of the triangles. This point also is equidistant from the midpoints of the sides of the triangle. In addition, the segment formed by the center and the midpoint of a leg is perpendicular to that leg of the triangle. Obviously, there are many relationships and patterns which can be discovered regarding the four centers of a triangle.