>According to the web page referenced above, the triangle
>should have sides with a length of 6.7664, which is
>different from the answer I got (6.4308). Assuming the web
>page is correct, could you please point out the flaw in my
>approach to the problem?

Well, there's one at the beginning:

> If the point of intersection is shifted from inside the
> triangle to the mid-point of one of the sides,

This is an entirely different problem.

> it can be said that the same three line segments (a, b, c)

The same?

> extending from the vertices now have lengths
> where a = h (height of the triangle), b = s/2
>(half the side length), and c = s/2. Since the
> sum of the line segments is fixed at 12, h + s = 12.

How is it fixed? It does not make sense to call a constant a fixed number. For the latter, something must be changing, while the fixed number maintains its value. What is changing in the original problem?

> Given an equilateral triangle, it is possible
> to express the height in terms of the side
> length in the following way.

Absolutely. But it has nothing to do with the problem.