Trisection of an angle using straight edge and compass only

[Hiten]

According to mathematicians, the trisection of an angle using a straight edge an a compass is an impossibility. It should supposedly give a cubic equation which cannot be solved using the given tools. However I think that I might have solved it. My maths teacher told me that he couldn't find a flaw in my theory. So I just wanted to know if what I have done is correct or no. And if it's incorrect then where have I gone wrong? The steps are given below:
Steps:
1) Create any arbitrary angle ‘θ’.
2) Label the point where it’s arms meet as ‘o’.
3) Create a circle with a radius ‘r’ with the centre ‘o’. This is circle 1.
4) Trisect the radius ‘r’. Twice of the trisected radius gives ‘R’.
5) Copy the arbitrary angle created in step 1 and label the point where it’s arms meet as ‘O’.
6) Create a circle with radius ‘R’ with it’s centre as ‘O’. This is circle 2.
7) Take the length of the chord in circle 2 which creates the arbitrary angle and make a chord in circle 1 of the same length.
The length of a chord is directly proportional to the arc that forms it.
9) Length of an arc= (θ/180)*r
10) The ratio of the arcs formed by the arbitrary angle in circle 1 and circle 2= ((θ/180)*r)/((θ/180)*R)=r/R .
But R=2/3r
∴ ratio=r/(2/3)r
∴ratio=3:2
11) By the properties of circle that chords make angles on the centre in ratio to the length of their sides:
The second chord that was made, should make an angle which is equal to two-thirds of the angle formed by the original chord.


Thus the angle is divided into two parts:
1) Two-thirds of the original angle
2) One-third of the original angle


Hence the angle is trisected.

[dcnick96]

Hi, Hiten. Our apologies for the delayed response to your question. I commend you for not accepting something on faith and seeing if you can find an alternative solution!

So, this is an age-old question dating back to the ancient Greeks. It is true (proven in the early 1800s) that you cannot trisect an angle using an unmarked straight edge and compass. Meaning, you cannot trisect an angle without measuring lengths, as you do in your procedure, or use other tools. I reviewed your steps; and I cannot find anything wrong with your calculations. I found several articles on the web outlining different methods to trisect an angle using a marked straight edge and compass.

Congratulations! I think this would be a great presentation for your Science Fair, since you developed a unique method. Be sure to do thorough research across several resources to ensure your method is not already published. In a very brief search, I didn't find anything. If you choose to present, perhaps outline why the original problem was mathematically proven impossible and discuss other published methods using tools other than an unmarked straight edge and compass. Finally, present your method.

Don't forget to mention that there are, in fact, two values of theta (the angle) for which you CAN trisect using only a straight edge and compass. I'll let you research and find those angles.

I hope this helps. Good luck; and keep searching for your own answers, even when others tell you it is impossible!