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Originally posted by Lew Hartswick View PostWas 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 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.
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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.Regards, Marv
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Originally posted by mklotz View PostThe 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.
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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.Regards, Marv
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Originally posted by mklotz View PostI'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.
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Originally posted by nickelcityfab View Post
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...Regards, Marv
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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 x1Qx4 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 = x1x2
b = x2x3
c = x3x4
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.
Regards, Marv
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