Erin
Mueller
Altitudes
and Orthocenter, A Little Proof
Theorem:
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:
1) 
From the formula for the area of a triangle, 
, we can use segment lengths from the picture above to
represent base and height. Therefore, 
.
AndÉ
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.