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
Algebra.Com
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) (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: \ =\ \cos(60^\circ)\ =\ \frac{1}{2})
If 
could be trisected, the minimal polynomial of )
over would be a second order polynomial.
Note the trigonometric identity: \ =\ 4\cos^3(\theta)\ -\ \3cos(\theta))
Let )
\ =\ \frac{1}{2}\ =\ 4y^3\ -\ 3y)
^3\ -\ 3(2y)\ -\ 1\ =\ 0)
Let 
Let \ =\ x^3\ -\ 3x\ -\ 1\ =\ 0)
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:
\ =\ 1\ -\ 3\ -\ 1\ =\ -3)
\ =\ -1\ +\ 3\ -\ 1\ =\ 1)
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 \ =\ 4t^3\ -\ 3t\ -\ \cos(\theta))
is reducible over the field extension \right))
John
My calculator said it, I believe it, that settles it
RELATED QUESTIONS
I have angle A which is an acute angle. The measure of the. Angle is not specified nor... (answered by solver91311)
If three times the supplement of an angle is subtracted from seven times the complement... (answered by addingup)
Given: Angle HKJ is a straight angle, KJ bisects angle HJK.
Prove: Angle IJK is a right (answered by sheldonbbtrocks)
if three times the supplement of an angle is subtracted from seven times the complement... (answered by nyc_function)
If three times the supplement of an angle is subtracted from seven times the complement... (answered by edjones)
Mateo is constructing an equilateral triangle inscribed in a circle with center P. He... (answered by mathmate)
) Use the fact that every point on the perpendicular bisector of a line segment is... (answered by solver91311)
Ok, there is a picture of a triangle ABC with a straight perpendicular line. The diagram... (answered by astromathman)
Is there a way, just using geometry (i.e. straight-edge & dividers) to find a square... (answered by richard1234)