Since our middle school or high school years, we have learned that by using only a compass, which can only transfer a known length, and a straight edge, which can only draw a straight line, one can not trisect a given angle. This old truth was found in the early years of Greek geometry, but it is only in the 19th century that people first gave it a solid proof. There are other similar “impossibility theorems” , such as squaring a circle, doubling a cube. However, you may wonder how to prove something is impossible, if you cannot or have not enumerate all possible methods on the first place.
Here, I will give you a comprehensive proof of this impossibility based on a fundamental algebraic approach. Modern mathematicians often give a more rigorous and concise proof based on Ring Theory, which is unluckily beyond of a high school student’s knowledge scope. While this one, I only use basic algebraic math that every average person can handle easily. So, it’s novice-PROOF!