No mathematical problem generates more "crankiness" than the trisection problem. Underwood Dudley coined the term "mathematical cranks" to describe people who somehow believe that they have discovered a "proof" of a math problem that has already been proven impossible. The first complete proof that it is impossible to trisect an arbitrary angle appeared in the early nineteenth century yet there are people who refuse to believe it. Dudley wrote a book called "The Trisectors" and published by the Mathematical Association of America where he described some of the more unusual and determined people who were convinced that they did indeed find a way to trisect an arbitrary angle.
In this book Yates does a superb job in describing the problem and the historical search for a resolution. The problem was originally formulated by the ancient Greeks and Yates starts there and engages in a gentle walk through history touching on the high points of the problem and the points of resolution along the way. Written at a level so that middle and high school students can understand it, he also covers mechanical devices that can be used and several methods of approximation.
This is a book that should be studied by all high school teachers of geometry. The trisection problem is one that can be understood by high school students with the proper presentation. Using this book, that obstacle will be overcome.