I was given this same problem by my dad when I was young. He said the same thing, that it was 'proven' impossible. I invested quite a few hours trying different ways, because it seemed intuitively wrong. I have disagreed with the conclusion of math. The 'proof' is in error:
The angle of any circle can be tri-sected and bi-sected to within the precision of the angle, the compass, and the ruler. I am afraid this is where this world and the math depart. Math is a language within the world, made to describe the world, and the language is fallible in that regard. The math is producing a FALSE statement. It IS possible to tri-sect an angle using a ruler and a compass to any precision desired with the correspondingly available precise tools. The procedure is this:
1. Bi-sect the angle in half. angle(Ray 1) = angle(X) / 2
2. Bi-sect the angle in half again. angle(Ray 2) = angle(Ray 1) / 2
3. Bi-sect the angle between step 1 and 2. angle(Ray 3) = (angle (Ray 1) + angle (Ray 2)) / 2
4. Repeat step 3 until the desired accuracy is reached.
The resulting sequence is:
Angle = 1/3 angle(X) = angle(X) * (1/2 -1/4 +1/8 -1/16 +1/32 -1/64 +1/128 -1/256 +1/512 -1/1024, etc….)
Where is the math error? The mathematician errors by extrapolating or interpolating what appears to be true at sizes that people are familiar with, to the sizes on the quantum and allegedly infinite precision.
The mathematician may note that it is impossible to tri-sect the angle with infinite precision. Infinite precision impossible? That is true. However, it is also impossible to bi-sect any angle with infinite precision. Furthermore, there is no ruler with infinite precision, and there is no compass with infinite precision, and if there were it would take an infinite length of time and an infinite amount of energy to carry out the bi-section… just as it would take an infinite length of time to carry out the tri-section to an infinite precision.
The mathematician may note that the math is considering a perfect circle, with a perfect compass, and a perfect ruler. Which is it… is the world imperfect, or is the language of math imperfect? It is impossible to produce a perfect circle, because there is no such thing as a perfect compass, and there is no such thing as a perfect ruler, in this perfect world.
The angle of any circle can be bi-sected, and tri-sected, to within the precision of the angle, any compass, and any ruler. At some level, math becomes a language for fiction. Either it is true to say that it is possible to tri-sect an angle with a ruler and compass, or it is false to say that it is possible to bi-sect an angle with a ruler and compass. Math wrongly suggests otherwise.
Perhaps another example of this fallibility of math will help. A distance is divided into two. Divide the resulting distance into two again. Repeat. The math says that the distance is always greater than zero. Prove it… DO IT. If you have not done it, then you have not proven it. The quantum physicist who has tried these things will report that it is impossible to measure the distance between two objects without affecting their position. The step ‘repeat’ at some level of precision is impossible. Impossible. Furthermore, the forces between the particles at close distances increases so that the proposed steps rapidly become impossible and imprecise. Furthermore, any known quantity, anything measured, any trait, cannot be divided as the mathematician suggests. There is a quantum limit.