Trisecting a Line

Date: 01/25/98 at 09:37:40
From: Led
Subject: Trisecting a line

Hello,

I've been researching, and it seems it is impossible to trisect an 
angle. But how about trisecting a line? Using propositions 1 - 34, 
Book 1 of Euclid's elements.

Thanks!

Led

Date: 01/25/98 at 10:27:50
From: Doctor Luis
Subject: Re: Trisecting a line

That's easy. You can use the properties of similar triangles to think 
of a way of trisecting a line. 

Take a line segment AB, for example (this is the segment you want to 
trisect). Next, take a line passing through point A that is at an 
angle to line AB (call it AC). Construct AC as a "multiple" of some 
arbitrary length (in this case it would be 3). This way, you know that 
line AC has been trisected.

               C
              /
             /
          P'/
           /
          /
       P /
        /
       /
      /
    A --------------------------- B
              Q        Q'

If you form a triangle by connecting points C and B, you'll get a 
line CB. Now make two lines parallel to side CB that pass through 
points P and P', respectively (these are the points that trisect AC). 
The two parallel lines will intersect AB at points Q and Q'. It is not 
hard to show that, indeed, points Q and Q' trisect AB.

What we've done is, basically, trisected a general line segment by 
constructing a line segment that we know is already trisected.

Informally speaking, we just divided up a line segment. That's an
interesting operation we can do with lines. See if you can come up 
with the equivalent of other arithmetical operations.

Just as we divided the line, we could as easily (well, maybe not 
just as easily, but it can be done) multiply and divide line 
segments (we can certainly add them and subtract them - THAT'S easy) 
and also take square roots. Indeed, there's a whole class of numbers 
that can be constructed (in the euclidean sense) with just a straight 
edge and a compass, and they are called algebraic numbers. It turns 
out, however, that there are some numbers that are not algebraic, 
and these mathematicians have called transcendental numbers, which, 
incidentally, is why we know that trisecting the angle with a straight 
edge and compass is impossible, since it is the equivalent of solving 
an equation with transcendental roots (transcendental numbers can't be 
constructed).

Examples of transcendental numbers? pi is a good example (pi is also 
irrational). The famous e is also transcendental. In fact, there are 
more real numbers that are transcendental than algebraic real 
numbers... (although there are as many algebraic numbers as there are 
integers).

Just some food for thought.

Doctor Luis,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/26/98 at 06:43:19
From: Led
Subject: Re: Trisecting a line

Dr. Math,

Thanks for the quick response. I'm really grateful for your help. 
I found my own way of doing it, because I can't use your solution 
since we also have to prove our solutions and constructions with only 
the propositions (or theorems) we have discussed in class. We can't 
just submit a solution, we also have to prove why and how this 
solution works. And we are studying geometry according to the order of 
the propositions of "The Elements of Euclid." We have only taken up 
Book I, Propositions 1 - 34. That doesn't include similar triangles 
yet, so I can't use those propositions.

Well, thanks again.

Led