Conchoid of Nicomedes

Stu put up a post a few days ago about trisecting an angle using standard geometric drawing techniques. I looked at it and wondered if I could validate this technique using algebra. So I dinked around for awhile, but didn't come up with a solution. Well, surely I can validate it using trigonometry, and I did. I put the calculations in a spreadsheet, and to the limits of Google's math capabilities, it does appear to be correct.

But this is really a simple looking problem. We just have three triangles, surely I can solve this problem using algebra. So I fiddled with it for another day, and eventually I did come up with something, but it has an exponent of 4, and there is no nice way to solve an algebraic equation like that.

So I have a solution of sorts, it will still require a computer to work out the actual numbers, but it doesn't require any trig. Let's see what we can find on the net. First stop is Mathematica, where they use two constants, but don't tell you where they fit in this picture. Second stop is Wikipedia, where I find this line:

"They are called conchoids because the shape of their outer branches resembles conch shells."

As for the trisection, I did pick up one insight. As angle BAC approaches 90 degrees:

• line segment j goes to zero,
• line segment AD goes to 2, and
• line segment a goes to 1,

so now triangle CAD is half an equilateral triangle, and angle BAD is 30 degrees, which is indeed 1/3 of 90.

Update May 2015: Replaced the missing diagram. Updated the quote from Wikipedia and added a link to their article. Removed the kibitzing about the original quote. Fixed a typo. Replaced the link to Stu's article.