Amberly Roberts

Assignment 8: Orthocenters and Altitudes

The following set of numbered steps will develop a constuction that we will investigate.

1. Construct any triangle ABC.

2. Construct the Orthocenter H of triangle ABC.

3. Construct the Orthocenter of triangle HBC.

4. Construct the Orthocenter of triangle HAB.

5. Construct the Orthocenter of triangle HAC.

6. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.

Click here to open a GSP file of this construction. This will allow you to move points A, B, and C and observe changes in the picture.

Observation allows us to formulate the following conjectures:

The triangle formed by the circumcenters of the inner triangles forms a triangle that is congruent to the original triangle ABC.

Each of the 4 circumcircles are congruent.

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