Othocenters and Corresponding Triangles
by Venessa Brown
The goal of this assignment is to construct an acute triangle ABC and it's orthocenter. Proving the following:
Construction: 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.
A. Prove that,
In order to prove this prpportion, I will make a comparison and use of the areas of three distinct triangles with triangle ABC to Triangle ABC itself.
Notice that the area of these three triangles AHC, BHC, and BHA equal to the sum of the area of Triangle ABC.
So we can say:
B. Prove that,
Notice that we can rewrite the numerators of the first proof in terms of the numerators of the proof we need here. Therefore we will start our proof using what we have proven in part A.some
HD = AD - AH
HE = BE - BH
HF = CF - CH
Then,
C. What if ABC is an obtuse triangle?
If ABC is an obtuse then the equations fail because CF =CH and BF = BH