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Trisecting an Angle: Proof


Date: 6/3/96 at 20:47:09
From: Michael Bronson
Subject: Tri-section

Dr. Math:

I have been out of touch with geometry for years, but remember 
trying with a couple other friends in grade 8 to trisect an angle, 
and prove it. 

My Question: Has anyone been able to trisect an angle and prove it 
with basic Euclidean geometry? 

Thank you.
Respectfully,
Michael Bronson


Date: 6/4/96 at 17:47:33 
From: Dr. Jodi
 Subject: Re: Tri-section

Hi Michael! 

You can trisect an angle but not with a ruler and compass Here's 
what I found at the Sci.Math.FAQ 

http://daisy.uwaterloo.ca/~alopez-o/math-faq/    

See the webpage for the picture...

(By the way, this archive also has info on a lot of interesting 
problems - most of them very accessible with no more than high 
school math. Good luck reviving your interest and refreshing your 
brain!) 

The Trisection of an Angle

Theorem 4. The trisection of the angle by an unmarked ruler and 
compass alone is in general not possible.

This problem, together with Doubling the Cube, Constructing the 
regular Heptagon and Squaring the Circle were posed by the 
Greeks in antiquity, and remained open until modern times. 

The solution to all of them is rather inelegant from a geometric 
perspective. No geometric proof has been offered [check?], 
however, a very clever solution was found using fairly basic results 
from extension fields and modern algebra.

It turns out that trisecting the angle is equivalent to solving a cubic 
equation. Constructions with ruler and compass may only compute 
the solution of a limited set of such equations, even when restricted 
to integer coefficients. In particular, the equation for theta = 60 
degrees cannot be solved by ruler and compass and thus the 
trisection of the angle is not possible. 

It is possible to trisect an angle using a compass and a ruler marked 
in 2 places. 

Suppose X is a point on the unit circle such that angle XOE is the 
angle we would like to ``trisect''. Draw a line AX through a point 
A on the x-axis such that |AB| = 1 (which is the same as the radius 
of the circle), where B is the intersection-point of the line AX with 
the circle. 

Figure 7.1: Trisection of the Angle with a marked ruler 

Let theta be angle BAO. Then angle BOA = theta , and angle XBO 
= angle BXO = 2 theta 

Since the sum of the internal angles of a triangle equals pi radians 
(180 degrees) we have angle XBO + angle BXO + angle BOX = pi 
, implying 4 theta + angle BOX = pi . Also, we have that angle 
AOB + angle BOX + angle XOE = pi , implying theta + angle BOX 
+ angle XOE = pi . Since both quantities are equal to pi we obtain

4 theta + angle BOX = theta + angle BOX + angle XOE 

>From which

3 theta = angle XOE

follows. QED.

-Doctor Jodi, The Math Forum

    
Associated Topics:
High School Constructions
High School Euclidean/Plane Geometry
High School Geometry

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