Triangles and Orthocenters

by

Hieu Huy Nguyen

Our goal in this investigation is to observe the behavior of triangles and orthocenters when a single vertex is displace in different positions.

We begin with the construction of triangle ABC with orthocenter H located within the triangle. We then construct 3 interior triangles using 2 vertices from traingle ABC and the orthocenter H giving us triangels HAB, HBC, HCA. Then we can generate circumcircles using vertices of each of the 4 triangles generated. Each of the circumcircles contain a circumcenter generated from using GSP script tools. The result of the construction is displayed below.

We can now begin to explore the behevior of the figure as it experiences changes in any vertex. Since the construction is a triangle, we understand that the behavioral affects are the same for vertex A,B, or C.

1) What happens when the vertx C is relocated to form and equalateral traingle ABC?

The results give us a figure in which the orthocener is located on the same point as the circumcenter of the circumcircle formed by triangle ABC. We also observe that a regular hexagon is formed from the connection of vertices of triangle ABC and the circumcenters of the circumcircles generated from triangles HAC, HAB, and HBC. Mathematically, this indicates that all four circles have and equal radius, which means they are all equal in size.

2) What happens when we move vertex C on to the same location as orthocenter H?

The results gives us an isoscoles triangle with points C and H located at the vertex of the equal sides. We can conclude that there is a single isoscoles triangle ABC in which there is no defined orthocenter H. We also notice that the circumcircle of traingle ABC is overlapping the circumcle of traingle HAB.

3) What happens if vertex C is located in a position which moves orthocenter H outside triangle ABC?

The results of the movement sends point C inside traingle HAB, and C becomes the orthocenter of triangel HAB. Bassically, the orthocenter H and vertex C changed positions from the original configuration. We can conclude that any verteice configuration which sends an orthocenter outside triangle would result in a new triangle where the original vertex becomes the new orthocenter, and the original orthocenter becomes the new vertex.

4) What happens if we keep the distance of AB the same and increas the angle of vertex C?

The results would increase the distance of point H (the original orthocenter) farther away create a traingle HAB with higher altitude. We can conclude that the altitude of a traingle increases as the orthocenter moves closer to the opposite side of the vertex.