Assignment 8

 

Page Bird

 

Altitudes and Orthocenters

 

12. Given triangle ABC. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully. Prove:


 

 


 

 


Click HERE for a GSP sketch. What if ABC is an obtuse triangle?

 


 

 

 


One way to approach this problem is to think of the area of triangle ABC as a sum of the area of the three triangles above which can be recognized with the three colors. 

 


So the area of triangle ABC can be written as:

 

 



Also, the area of triangle ABC divided by itself is 1

 

 


There are several ways to express the area of triangle ABC.


 

 


Another way to express the area of triangle ABC divided by the area of triangle ABC

is


 


 

The goal is to reduce this.  The first step in the process is


 


This can be reduced to


 

 

 


Since

 

1/2CF(AB) = 1/2AD(BC)

and

1/2AD(BC) =1/2 AC(BE)

 

We can substitute into the equation above.  Therefore,


 


This reduces to

 


 


Therefore this proves our statement through a series of substitutions and simple geometric relationships.

 

 

Next, we want to show that


 

Let’s go back to the picture:

 


 

 

 

 


 Looking at the picture above, we see that

AH = AD – HD

BH = BE – HE

CH = CF – HF

 

These can be substituted into the fractions to get


 


This can be rewritten as


 


And then reduced to


 

 


Since


 

 


Then 3-1=2.

 

Therefore,

 


 

 



Now this brings us to the question, does this apply to obtuse triangles?

 


To find out, I constructed the orthocenter for an obtuse triangle.  The orthocenter lies outside the triangle when it is obtuse.

 

 

 


The red triangle represents the given obtuse triangle, and H is the orthocenter.  I then found that the orthocenter of the triangle BCH was point A.  Based on this, I measured segments HF, CF, HE, BE, HD, and AD to see if the relationship held.


 

 


They did not.  Based on this and the fact that the orthocenter lies outside the obtuse triangle, this relationship does not hold for obtuse triangles.

 

To check this out yourself, click here.

 

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