Assignment 8
by
Allison McNeece
Altitudes and Orthocenters
In this first part we will look at a triangle, its orthocenter, the triangles formed inside the larger triangle with the orthocenter as a common vertex and, lastly, the othocenters of each of these smaller triangles.
1. Construct any triangle ABC 

2. Construct the Orthocenter H of triangle ABC 

3.Construct the Othrocenter of triangle HBC Notice that this is also the vertex A of the triangle ABC 

4. Construct the Orthocenter of triangle HAB Notice this is also the vertex C of the triangle ABC 

5. Construct the Orthocenter of triangle HAC Notice this is also the vertex B of the triangle ABC 
Next we will find the circumenters of each of the triangle and form their circumcircles:
1. Find the circumcenter of ABC and construct the circumcircle 

2. Find the circumcenter of HBC and construct the circumcircle 

3. Find the circumcenter of HAB and construct the circumcircle 

4. Find the circumcenter of HAC and construct the circumcircle 

5. All together now 
Do you see anything interesting when we put all of the circumcircles together?
Below is the figure for you to play with. You can click on the vertices A,B and C. Move them around and see how the circumcircles change.
What happens if you move one of the vertices to touch the orthocenter H?
What if you move the vertices such that H is outside of ABC?
Let's play around a bit more with these circumcircles:
Let:
circumcenter of HAB = a
circumcenter of HBC = b
circumcenter of HAC = c
circumcenter of ABC =d
What does the triangle of abc look like?
What is the orthocenter of abc?
oh snap! that's the circumcenter of ABC!
Hmmmmm... What about the circumcenter of abc?
Well, that's just the orthocenter of ABC!