More explorations about Orthocenter
by Morgan Guest
Given triangle ABC, construct the orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C, respectively.
Click here for a GSP sketch of the orthocenter.
First, we will prove the first equation. To add these fractions, we need a common denominator. I will find the common denominator by exploring with the ratio of the areas of three triangles.
Below is an image of the 3 triangles:
There are 3 pairs of similar triangles and we will explore each of them in order to find a common denominator. Keep in mind that the area of a triangle is (1/2)(base)(height).
The first pair of similar triangles are triangle ABC and triangle HBC. The ratio of the areas of these triangles are:
The second pair of similar triangles are triangle ABC and triangle AHC. The ratio of the areas of these triangles are:
The third pair of similar triangles are triangle ABC and triangle ABH. The ratio of the areas of these triangles are:
We can simplify all three of these equations:
Using the new equations, we now can write the original equation with a common denominator:
Finally, we can combine the ratios and we have proved the first equation and we can see in the following pictures that this equation makes sense visually.
Now we can prove the second equation by using the first equation:
We need to rearrange how we define the segments by representing it with the entire altitude of the triangle minus the matching piece.
HD=AD-AH, HE=BE-BH, and HF=CF-CH
Now plug the rearranged formulas into the previously proven equation:
Now we have proved both equations.