Assignment 8

Altitudes and Orthocenters

Kate Berryman

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Altitudes and Orthocenters

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.



We can start with the formula for the area of a triangle.


In our case:

So we can have:

We also know that:


If we look at our smaller triangles, we see that:

So another way to calculate area is:


We recall:

Then we divid by AD*BC:



We can substitute our denominators:

Therefore, we arrive at:


Now let's prove:

We proved above that the follow is true:

Now let's look at our triangle.


We can easily see that:


So by a little substitution we get the following:



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