Altitudes and Orthocenters
By
Jeffrey Frye
The construction for this assignment is shown below. The orthocenter for triangle ABC has been constructed and is labeled H. Circumcircles have been constructed for each of the triangles formed that include the orthocenter as a vertex and for the original triangle.
The circles are all the same size and remain the same even when the shape of the triangle is changed and the size of the circles change. When the orthocenter, H, is moved to change place with the vertex, this relationship remains the same. This is true since the movement is along the altitude of that vertex, and it only changes the intersection point for the other two altitudes to outside the triangle. This is what is happening for an obtuse triangle. The circles become larger the further away the orthocenter is from the vertex of the original triangle.
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