Erin Mueller

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.


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