John the shortest distance between two points on a flat surface is a straight line.
Show this is the case, presuming you mean any flat surface, and I'll gladly admit I'm wrong. Also, why it is relevant.
The triangle inequality, a <= b+c is true in flat space.
Consider the straight line AB and some curve c that goes from A to B and isn't equal to AB.
A set of C points will be used to approximate c.
Step 1
Pick a point C1 on c, the path A,C1, B is clearly longer than AB by the triangle inequality.
Step n
Pick another point Cn on the path c, then find the points Cr and Cs that are before and after Cn on the curve c.
The path Cr,Cn,Cs is longer or equal to the path Cr,Cs by the triangle inequality. So the approximation of c given by the n C points is still longer than AB.
So by induction any approximation of a curve from A to B by a finite number of points is longer than AB.
This can be extended to the length of the curve itself.
Consider the sequence T. The value of Tn is the length of a finite approximation of c by a set of points where no two successive points are more than 1/n distance apart. Also the approximation used for Tn+1 must contain all the points used for Tn.
The limit of the sequence will be greater than AB, and will be equal to the length of c.
Therefore the straight line AB is the shortest path between A and B.
I feel like my proof has been ignored.
I missed it.
I'm sorry, you seem to be using some terms you haven't defined, such as some straight line AB. How exactly do you define a straight line? And such a straight line that lies from A to B?
I just meant the normal definition of a straight line, it's a primitive object of the geometry and any two points share a line. Come to think of it straight lines are normally given the property of being the shortest path between two set points, so my proof it a bit redundant if that is an axiom.
I have a feeling the definition of straight line is different in the geometry you have, so I think it would be easier if you give the axioms of it and I see if I can still find a proof.
Small question, what is it that makes you geometry flat instead of +/- curved?
So then you can understand why I would be skeptical of any proof that relies upon an axiom that directly states the intended proof. Likewise, had you defined it along the lines of "a set of points that lie evenly" you might find issue when dealing with the matter of curves.
The Earth is Flat. The issue is our idea of space is incorrect. This can be seen by considering the path of a perfectly orbiting satellite. Since we know it is traveling a straight line, it can be then seen easily that the surface of the earth also follows a straight path. You can find a bit more about it by considering this brief thought experiment:
http://www.theflatearthsociety.org/home/index.php/blog/einsteins-relativity-proves-earth-flatAs those dragon enthusiasts out there, they have got you thinking so little that they have told you that giant impossible beasts existed and you just swallow it whole with little thought. Whenever a legitimate attack against these mythical beasts lands a decisive blow, the goal post is moved.
Oh yeah, their bodies are sustainable enough - if you fill all their bones with air. Its about as silly as the childhood joke of filling a bunch of balloons up with helium and soaring up to the sky. What's that? their size would have caused them to deplete the nearby food sources in almost no time? Of course not, you know.
Because dragons.
Their head would give them explosive heart pressure? I think you missed the bit where everybody has decided to believe in this nonsense from the get go because some rich old bastard faked a skeleton for his entertaining lobby. Then comes the wanna-be that started digging out crocodile teeth and bones and stitching them together with Iguanas. And this isn't even getting to the Bone Wars - science has never known a time of less honesty and more fakery. But yeah sure. Giant flying impossible lizards with bones filled up with air, a magic food source, and blood pressure that would make any living creatures heart and circulatory system explode.
Dragons sound totally legit.