SOLUTION: I understand that "trisecting a given angle" is under compass -and -straight- edge- rules impossible. Is there a geometric proof that proves it can not be done? Please share with

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Question 332688: I understand that "trisecting a given angle" is under compass -and -straight- edge- rules impossible. Is there a geometric proof that proves it can not be done? Please share with me.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


There is not a geometric proof of which I am aware. You have to rely on algebra and trigonometry. The following courtesy of Wikipedia:

Let denote the rational numbers.

A number constructible in one step from field K is a solution of a second order polynomial. Also, radians is constructible.

But radians cannot be trisected.

Note:

If could be trisected, the minimal polynomial of over would be a second order polynomial.

Note the trigonometric identity:

Let









Let

Let

The minimal polynomial for (hence the minimal polynomial for ) is a factor of .

If has a rational root by the Rational Root Theorem, that root must be either . But:





Hence is irreducible over and the minimal polynomial for is of degree 3.

Therefore cannot be trisected.

In fact there are some angles that can be trisected. But an angle can be trisected if and only if is reducible over the field extension

John

My calculator said it, I believe it, that settles it