EMAT 6680 :: Clay
Kitchings :: Assignment 8

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:

Part 1:

Part 2:

Part
1: Let us first set out to investigate
how the ratios could possible sum to 1 in the first case.

To
begin, we will see if we can find a convenient way to write these ratios in
some other form, preferably a form that has common denominators.

I
have color-coded smaller triangle within triangle ABC in order to help
illustrate the method behind the Òmadness.Ó We shall consider the ratio of the areas of all the three
colored triangles in relation to the area of triangle ABC.

Consider
the following ratios :

**

Notice
that I did not use the same expression for the area of triangle ABC each
time. For the area of (ABC) we
have some options to use. It was convenient for each denominator in each ratio
to have an expression for the area of (ABC) such that one of the factors would
divide out with a factor in the numerator.

Now,
let us consider the sum of the following ratios:

(Common denominators)

Now,
the sum of the three smaller (colored) triangles in the numerator sum to the
entire area (ABC). Therefore we
can substitute our ratios we found earlier into the equation above:

by substitution ** from above.

Part 1: QED

We want to verify that:

Let
us rewrite the numerators knowing the following:

HD = AD – AH, HE = BE – BH, and HF = CF – CH

(The above is allowed due
to basic segment addition and subtraction.)

We
will replace the numerators in the previous proof with the expressions for HD,
HE, and HF above and simplify to derive the proof:

QED