I could not understand the math or the language to “prove” that the top line “C” in the diagram was a straight line. But, it’s pretty obvious to me that it is a straight line. But, since it is “proof” that is wanted and not an argument in philosophy about subjective reality (two parallel lines looked at from certain angles will eventually look straight as well from a subjective point of view), what I wonder is if the alternative method provided below would “prove” that “C” is a straight line.
Using the diagram provided as instructions, what if I built a small model out of wood? Then, I could put a carpenter’s level on top of “C” and tilt it until the tool reads that it is level; hence straight. Or, perhaps, I could even more simply use a straight edge (such as a ruler) to rest along the top of “C”. Wouldn’t that empirically prove that “C” is straight? I realize that my methods are not geometric equations, but wouldn’t they still be proof? If all else failed, couldn’t I remove the top line “C”, take it to a mathematician, and ask him or her if it was straight, and then see if they could figure it out without all of the extra agony? Carpentry sounds easier and more fun than geometry but it’s based on the same principles, I think, but since I’m not a mathematician I’m not sure.
Why couldn’t I put a level on a pile of cow dung and tilit it until it reads level? Would that prove the cow dung was straight? Level and straight are unrelated.
Just one of several problems I see with your approach.
In the real world, an absolutely straight line is impossible to replicate. Your carpenter’s level or straight edge is not straight, or at least it is impossible to know that it is straight. It could be .0000000001 mm. off of a straight line and be undetectable to the human eye, but still not straight.
It would prove that line DCE is straight for that particular triangle. So all you’d have to do is build a model of every possible triangle (allow some time) and you’d be good to go.
The simple definition of a line can prove this math question. A line, or in this example a line segment is the shortest distance between two points. The shortest distance is always going to be straight. The line could be sloped or at an angle, but it will always be straight.
The diagram as given is only there to help you understand the proof. For a mathematician, a mathematical proof could be given purely in mathematical notation, but it can be explained in ordinary sentences containing some mathematical notation. It doesn’t depend on the diagram at all.
Indeed, a diagram may be misleading. If you say “any triangle”, then there are in fact an infinite number of diagrams that you might build. And just because a proposition has worked for the first billion diagrams that you tried out does not mean that it will work for diagram no. 1,000,000,001
I think you’ve misunderstood the problem. “Prove DCE is a straight line” means prove that the three points D, C, and E are co-linear. They wouldn’t have to be.
FWIW, chel, your instinct isn’t all wrong. The famous ancient mathemetician Archimedes first solved the problem of determining volume for spheres, cylinders and other three-dimensional round things by building and weighing them. The mathemeticians of his day were outraged. In that sense, what has been said here is true. In geometry, one looks for analytic proofs of universal application (or proofs that no such proof can be constructed). Perhaps more importantly, his was the context of the question as originally asked, and as answered by Cecil.
There’s nothing wrong with experimental mathematics, i.e., trying out a few examples and seeing what happens in those cases. However, what happens can only be suggestive: no matter how many examples you look at, you aren’t looking at a proof (unless you’re looking at a case where there is a known finite number of examples, e.g., the set of even prime numbers.
As an example of experimental mathematics, I once run a computer program to find the angles of triangles with integer sides. Of course, I knew that the 1-1-1 triangle has 60 degree angles, and that the 3-4-5 triangle has a 90 degree angle, but I was really surprised to find that the 3-5-7 triangle has a 120 degree angle. (I haven’t seen that result published anywhere, but I haven’t looked very hard, and I doubt if I’m the first to find such an easy thing). What’s even more interesting is that because of this property of the 3-5-7 triangle, the 3-7-8 and the 5-7-8 triangles each have a 60 degree angle – and then the 3-10-13 and the 5-11-13 triangles again have a 120 degree angle. Going on from this in the obvious way, there are an infinite number of triangles with mutually prime integer sides with 120 degree angles, and also another infinite set with 60 degree angles – just as there are an infinite number with 90 degree angles. A very interesting result, and one that I was led to by a mathematical experiment. (And though i discovered it myself, as I said above I’m sure I’m not the first to find it!)
Giles, your point is perfectly valid. And, describes “the Rest of the Story” (as Paul Harvey says) in the Archimedes anecdote I related. That is, knowing the answer he eventually developed a proper (albeit non-Euclidian) proof, as least to the extent of demonstrating the relative proportions of spheres and cylinders. For more information, see here and here.
I have laid out this problem in AutoCAD for ten random triangles ABC and each time DCE proved to be a straight line within .00000000 which is as good as you’re gonna get.