Trisecting a LineDate: 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 |
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