Triangle Problem

A number of years (about 31) ago, when I used to work at RTI in the Research Triangle Park, there was an organization called the Triangle Universities Computational Center. They used to put out a newsletter and, on the last page, they often had a nerdy little math problem. I'd often work through the problem because, well, I was a nerdy kind of guy.

One problem proved to be particularly difficult and, oddly enough, it was about triangles. Just for kicks, I'll give it to you.

Farmer Jones had a big field that had the shape of an equilateral triangle. When it came time to spread fertilizer, he needed to know how big his field was so that he would know how much to buy. Problem was, he didn't know the dimensions of his field. All he knew was that there was a natural spring inside the field and that it was 200, 300 and 400 ft from the corners of his field. So how big was Farmer Jones' field? Help him out so he won't waste money buying too much fertilizer.

Or, if you like, given an equilateral triangle with sides x, and internal distances a, b and c, what is the area of the triangle in terms of a, b and c?

I figured out the answer pretty quick using an iterative approach, where I assumed a value for x and calculated the internal angles, which have to add up to 360 degrees. You just vary x until the sum is 360. I even wrote a computer program to do the calculations for me. But I wasn't real happy with that. I wanted an analytical solution. It took me about two weeks of intensive work in the evening and on a couple of weekends before I came up with a pretty solution.

I've shown this problem to a lot of people and only one other person has figured it out. Maybe he was the only one to try. He was a guy I worked with who handled shipping and receiving but had formerly worked in surveying. Actually, one other person figured it out using the iterative approach. I told him that it wasn't a very elegant solution. He disagreed.

Anyhow, give it a try and let me know if you find a cool solution. I'll post an answer someday.