Proving Bisection of Angles with Euclidean Geometry

[StephenPrivitera]

I was at brunch this morning and I met a young man of about 35 years who is a musician but studied mathematics in college. He mentioned to me that one of the great problems of mathematics is the problem of trisecting the angle. He taught me how to bisect an angle, and kept emphasizing that this pertained to Euclidean geometry. His description was that "in Euclidean geometry, you get a straight edge and a compass." How would one be able to prove that an angle has been bisected using one these tools and without applying theorems? I figured there must be some theorems to go along with this description. I want to be clear. What is Euclidean geometry? What does it include? What kind of theorems would Euclidean geometry include? I thought Euclid worked with Cartesian planes. Would it be valid to use concepts like slope in Euclidean geometry?
I recognize this question is large in scope. Ideally, I would like to be referred to some book that could give me an in depth understanding of Euclid's work, but I would also like a quick description if you can.

 

[m-theory]

how could someone interested in mathematics switch to music!

 

[StephenPrivitera]

Some people enjoy more than just science and math, I suppose. I can't understand why!

 

[KLscilevothma]

What is Euclidean geometry? What does it include? What kind of theorems would Euclidean geometry include?

Euclidean geometry is the geometry that we learn in school, like angle sum of triangle is 180 degrees. It bases on mainly 5 postulates and some definations. The 5 postulates are:

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
Postulate 1.
To draw a straight line from any point to any point.

Postulate 2.
To produce a finite straight line continuously in a straight line.

Postulate 3.
To describe a circle with any center and radius.

Postulate 4.
That all right angles equal one another.

Postulate 5.
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

According to what we've learned in schools(before university), all the 5 postulates are true. In fact, postulate 5 isn't really correct if we draw "straight lines" on a ball/curved surface. How can we define a "straight line" on a curved surface? In that case, we need non-euclidean geometry.

There are more than 1 models of non-euclidean geometry, depends on what curved surfaces we have. In some case, angle sum of triangle is less than 180 degrees and parallel lines meet at some point on the curve, etc.

 

[Guybrush Threepwood]

It seems that for the ancient greeks the geometry problems had to be solved using compass and straightedge. I guess that your friend was referring to this when he said "Euclidean geometry" although these days it has a larger meaning...
Angle trisection is one of the problems the ancient greeks couldn't solve because of their way of thinking...
See some more problems like this

 

[KLscilevothma]

1. circle squaring.
2. cube duplication.
3. angle trisection.

To prove they are insolvable by only using compass and straightedge is rather complicated.

 

[HallsofIvy]

To answer your question about proving that the bisection of an angle works:

The standard method of bisecting an angle is: Place one end of a compass at the vertex of the angle and strike an arc. At the two points where that arc crosses the angle, strike two new arcs. Draw the line between the point where the two arcs cross and the vertex of the angle. That line bisects the angle.

To prove that, draw the lines from the point where the arcs cross to each point where your original arcs crossed the angle. The line between those two points and the other lines give two equilateral triangles. One then shows that the "bisecting" line cuts each of those into two congruent triangles and so the has cut the original angle into two congruent angles.

 

[HallsofIvy]

Saying "Angle trisection is one of the problems the ancient greeks couldn't solve because of their way of thinking..." is a bit simplistic and might be interpreted as saying the ancient Greeks just weren't thinking correctly!

Actually the problem the Greeks had was that they didn't have a good numeration system and, as a result, didn't have "algebra". They used geometry as a way of handling algebra (see Euclid's works on proportions, etc.)

In particular, numbers could be "constructed" by starting with a given unit length and then using compasses and straightedge to mark basic circles and straight lines (a "marked" straightedge was illegal because that required numbers given independently of the orgiginal unit).

It's trivial to show that any rational number is "constructible". The fact that the hypotenuse of a right triangle with legs of length 1 is [sqrt](2) shows that SOME irrational numbers are constructible.

It wasn't until the nineteenth century that it was shown (through abstract algebra) that constructible numbers are precisely those numbers that are "algebraic of order a power of 2" that it was proved that some numbers are NOT constructible.

In particular, if it were possible to trisect an angle, it would be possible to construct a number that is algebraic of order 3 (satisfies a cubic equation- that's from the fact that cos(3[theta]) can be written in terms of cos³([theta])).

If it were possible to "duplicate the cube" (construct a cube exactly twice the volume of a given cube- using "compasses" that strike a surface of a sphere and "straightedge" that draws a plane through three points), it would be possible to construct a segment of length cube root of 2, again, algebraic of order 3.

If it were possible to "square the circle" (construct a circle having exactly the same area as a given square), then it would be possible to construct a segment of length [sqrt]([pi]). It wasn't until the late nineteenth century that it was shown that [pi] is "transcendental" so neither it nor [sqrt]([pi]) is algebraic of any order.

 

[StephenPrivitera]

Originally posted by HallsofIvy
It wasn't until the nineteenth century that it was shown (through abstract algebra) that constructible numbers are precisely those numbers that are "algebraic of order a power of 2" that it was proved that some numbers are NOT constructible.

It wasn't until the late nineteenth century that it was shown that [pi] is "transcendental" so neither it nor [sqrt]([pi]) is algebraic of any order.

Can you elaborate a little bit on what you mean by "algebraic of order n?"

Where can I learn about nonEuclidean geometry?