Circle Division

[Paul Alciatore]

Well, don't let me stop you from showing the proof.

Don't mean to be disagreeable but he answer is yes. The Peano axioms and Zermelo-Frankel set theory provide a solid foundation for basic arithmetic. Even if one objects to set theory as a basis for arithmetic foundations, the Peano axioms can also be understood and established by way of category theory.

[Paul Alciatore]

And EXPLAINING it.

[Lew Hartswick]

Was told, as of 1949, by my Trig / Geometry teacher the "angle" couldn't be trisected. Has there been any new construction technique that does it???
...lew...

[nickel-city-fab]

Was told, as of 1949, by my Trig / Geometry teacher the "angle" couldn't be trisected. Has there been any new construction technique that does it???
...lew...

I've heard the same thing. So I actually went and downoladed Euclid's geometry, the original greek book has been translated. Link: https://archive.org/details/thirteenbookseu03heibgoog

I've wondered about simply drawing a "chord" across the end of an angle, and dividing the line into whatever number of parts such as 3 parts, like you said, or 7 parts. Instead of being able to do it directly. I think it would work.

[mklotz]

The statement is "trisection of an angle USING ONLY STRAIGHTEDGE AND COMPASS is not possible", not "trisection of an angle is not possible". AFAIK, the first statement still stands.

There are probably many ways to trisect an angle.

The trisection "tomahawk"...

https://en.wikipedia.org/wiki/Tomahawk_(geometry)

is an example of one trisection mechanism, although it cannot be used on certain angles.

[nickel-city-fab]

The statement is "trisection of an angle USING ONLY STRAIGHTEDGE AND COMPASS is not possible", not "trisection of an angle is not possible". AFAIK, the first statement still stands.

There are probably many ways to trisect an angle.

-- Indeed -- and the method I mentioned uses only a straightedge and a compass. The worked examples that i have here show "dividing a line into an arbitrary number of equal parts".

[mklotz]

I'll be looking forward to your demolition of a proof that's survived for 184 years. Be sure to publish it here for all to see.

[nickel-city-fab]

I'll be looking forward to your demolition of a proof that's survived for 184 years. Be sure to publish it here for all to see.

Notice how I said you draw a chord? making your angle into a triangle. You then divide the chord line into whatever (hell we did this in HS) and draw from those points back into the center of your circle...

[mklotz]

Notice how I said you draw a chord? making your angle into a triangle. You then divide the chord line into whatever (hell we did this in HS) and draw from those points back into the center of your circle...

I understand what you said. Now you need to show the math to prove that your method works. A verbal description of what you intend to do isn't enough.

[mklotz]

I think the attached diagram captures your proposed technique for trisecting an angle. If it doesn't, at least it will suggest a method for proving your technique.

I've turned things around for the proof. I've started with three equal angles (A in the figure). On the middle angle I've constructed a bisector (PQ length 's') and drawn the line x1-Q-x4 perpendicular to 's' that cuts across all three angles.

If your technique works, it should be possible to show mathematically that the angles divide the line into three segments...

a = x1-x2
b = x2-x3
c = x3-x4

that are equal, i.e. a = b = c.

Aside: To my eye, it already appears that 'a' is longer than 'b', but in math "eyeballing" carries no weight so on with the proof.

We can solve for 'b' in terms of 's'

b = 2 * s * tan (A/2)

We can also solve for:

z = Px2 = s / cos (A/2)

w = Px1 = s / cos (3A/2)

Knowing 'z' and 'w', we can use the law of cosines to find 'a'.

a = [w^2 + z^2 - 2 * w * z * cos(A)]^0.5

So, all you need to do now is prove that a = b using the expressions above. If you can then your technique will work. Good luck; be sure to show us your work.