Altitudes and Orthocenter, A Little Proof
Given triangle ABC, let H be the orthocenter and let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully. Prove:
If we let A1 = the area of triangle HAB, A2 = the area of triangle HAC, and A3 = the area of triangle HBC, then A = A1+A2+A3=area of triangle ABC.
Using properties of Algebra, we can conclude:
From the formula for the area of a triangle, , we can use segment lengths from the picture above to represent base and height. Therefore, .
Plugging in our areas for A, A1, A2, and A3 into the proportion above, we arrive at .
For the second part of the proof, we can see the following:
AH = AD – HD
BH = BE – HE
CH = CF – HF
Substituting this into we get
Now in the case when triangle ABC is obtuse, the results no longer hold since the orthocenter H lies outside of our triangle. But if we consider the triangle HBC, then our orthocenter is at A and the results shown above will hold.