Assignment 8

Altitudes and Orthocenters

Altitudes and Orthocenters

In this activity we will investigate properties of altitudes and orthocenters

Start by constructing any triangle ABC.

Construct the orthocenter H of triangle ABC. Remember the orthocenter of a triangle is the point of intersection of any two altitudes of the triangle. Notice that the orthocenter lies in the interior of triangle ABC, since ABC is an acute triangle.

Construct the orthocenter of triangle HAB. Remember that the orthocenter of HAB will lie outside the triangle since HAB is obtuse. Notice that the othocenter of HAB coincides with point C of the original triangle.

Now construct the orthocenter of triangle HBC. This time the orthocenter lies coincident with point A. It appears that when a triangle is constructed from two vertices and the orthocenter of an original triangle, the orthocenter of the constructed triangle lies coincident with the vertex of the original triangle not included in the constructed triangle.

Let's test the above observation by constructing the orthocenter of triangle HAC. Again, notice that the orthocenter of triangle HAC lies coincident with point B, the only vertex of the original triangle not included in the constructed triangle.

Now let's investigate the circumcircles of these four triangles. Construct the circumcircles of triangles ABC, HBC, HAB, and HAC.

Each of the
constructed circumcircles are congruent to one
another. Click here for an
interactive javasketchpad exploration. Drag each of the
vertices of the original triangle one at a time to coincide with
orthocenter H. Notice that each of the three circumcircles of the
smaller triangles are congruent to the circumcircle of triangle ABC.

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