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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
triangle ABC
2. Construct the Orthocenter H of triangle ABC
orthocenter H of ABC

3.Construct the Othrocenter of triangle HBC

Notice that this is also the vertex A of the triangle ABC

orthocenter of HBC

4. Construct the Orthocenter of triangle HAB

Notice this is also the vertex C of the triangle ABC

orthocenter of HAB

5. Construct the Orthocenter of triangle HAC

Notice this is also the vertex B of the triangle ABC

orthocenter of HAC

 


 

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
circumcircle of ABC
2. Find the circumcenter of HBC and construct the circumcircle
circumcircle of HBC
3. Find the circumcenter of HAB and construct the circumcircle
circumcircle of HAB
4. Find the circumcenter of HAC and construct the circumcircle
circumcircle of HAC
5. All together now
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.

 

Sorry, this page requires a Java-compatible web browser.

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?

triangle formed by abc

 

 

What is the orthocenter of abc?

orthocenter of abc

oh snap! that's the circumcenter of ABC!

 

 

Hmmmmm... What about the circumcenter of abc?

circumcenter of abc

Well, that's just the orthocenter of ABC!


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