Here is triangle ABC with the orthocenter (H) labels and the feet of the altitudes labeled as well.

Here is a GSP file for this triangle.

Part 1:

Prove .

First, we will look at the area of the large triangle, triangle ABC.

Therefore we have a relationship of the lengths of the sides to be

Now we will look at the area of three of the smaller triangles in regards to the larger triangle. We can say that:

(area of triangle ABC) = (area triangle BHC) + (area triangle AHB) + (area triangle CHA)

and

So we can set this formula equal to the area of triangle ABC and we get:

We can multiply the entire equation by 2 to cancel out the 1/2 so we are left with:

We are trying to solve for 1 so let's divide each side by . After we divide we are left with

Recall that we found at the beginning of the assignment that the relationship was true. We can use this to substitute in the denominator. We choose our substitutions so that we are left with the lengths AD, BE, and CF in the denominator (because these are the denominators in our equation we want to prove)

After substitution we get:

then,

so we are left with our desired equation of

Part 2:

Prove:

We well use our previous result in this proof. Based on the our constructed triangle we can use the following statements about segment addition:

Now we can substitute these equations into our result from Part 1:

We can can break up each fraction and simplify to get

By addition we get

If we move the 3 to the right side of the equation and the negative signs cancel each other out. So we are left with our desired equation of

.

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