by: Doris Santarone
Assignment #8: Properties of the Orthocenter
Consider triangle ABC, and its orthocenter H.
Goal #1: Prove 
Consider the area of triangle ABC.
From this, we can find that 
. We will use this later*.
Going back to the area of triangle ABC...we could consider this triangle as a sum of three smaller triangles, triangle BHC, triangle AHB, and triangle AHC:
Now, I will use the two equations 
and 
and substitute, giving me:
With some manipulation, I get:
From the equation* that I mentioned before, , we can substitute into the denominators of the 2nd and 3rd ratios, and we get:
Goal #2: Prove 
From before, I already know that . I also know, using segment addition, that:
Now, I will substitute:
What if the triangle is changed to an obtuse triangle? Consider the image below:
As you can see, 
, thus 
.
If and , then 
would have to be less than 0, and that is not possible.
So, we can conclude that the property does NOT generalize to obtuse triangles!
Similarly, you can see that 
, thus 
. And 
, thus 
.
If and , then 
.
If and , then 
would have to be less than 0, which is impossible.
So, we can conclude that the property does NOT generalize to obtuse triangles.
To see the GSP Sketch, click HERE. Make sure to move the triangle around to see that the properties remain true, except when the triangle is an obtuse triangle.