Assignment 8
by
Jackson Huckaby
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.
We will proceed by proving this in two separate parts. We will start with the first equation.
What this equation is suggesting is that the summation of the ratio of sides of 3 similar triangles is equal to one.
To add these fractions we need to first discover a common denominator. To do so we are going to look at the ratio of the areas of three triangles.
The first two triangles can be seen here:
We can see that our similar triangles are the triangles ABC and triangles BHC. To bring our side ratios in we are going to look at our area formulas.
Remembering that the area of a triangle is (1/2)(Base)(Height), we get
Our next triangles, triangle ABC and triangle AHC are seen here:
Our last pair of triangles are triangle ABH and triangle ABC.
If we simplify the right side of the equations we get that:
, 
, 
So now, instead of using our original equation:
We can rewrite this with a common denominator!
So now we can combine the ratios.
And now we can think about what we have stated. The note that the numerator of the left side is the summation of the areas of the three sub triangles, which is equal to the area of the entire triangle ABC. Therefore we can substitute and get:
.
Our next goal is to prove the equation:
We can now use our initial equation since it has been previously proved:
The first thing we want to do is provide a new way to represent the numerators.
HD=AD-AH, HE=BE-BH, and HF=CF-CH
What we have done is rewrite each segment by now representing it as the entire altitude of the triangle minus the matching piece.
We now substitute our new expression into our initial equation and get:
next we break up our common denominators:
then we simplify:
apply the commutative property:
factoring out a negative 1:
subtracting the 3 and dividing by -1:
!!!