Altitudes and Orthocenters

Rozina Essani

12. 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.

Prove:

Let us first draw
triangle ABC and the altitudes of the vertices. The altitude of A meets BC at D, B meets AC at E and C meets AB at F.
The point of intersection of the three altitudes is the orthocenter H.

We know that the area
of a triangle is (1/2) x base x height.

Area of ABC ===

Area(ABC) = area(BHC) + area (AHB) + area(AHC)

=== + +

1 =

Hence we have proved
the first part. Now lets look at the second part.

From the GSP
illustration we know that

AH = AD – HD

BH = BE – HE

CH = CF – HF

Now substituting into
our original equation we get

from the proof of the first part we have = 1

Hence .

Let us apply lengths to
the segments in triangle ABC and check to see if these equations hold.

Will the same apply to
an obtuse triangle?

As we
can see by changing to the triangle around to create obtuse triangles that each
figure is missing one of three components to the orthocenter. This will not allow for the equations to work on
obtuse triangles.