Shortest distance between two points. Line?

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dextercioby
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Suicidal said:
I would like to see the proof that the shortest distance between two points is a line. I found a proof online http://www.instant-analysis.com/Principles/straightline.htm but i can't quite follow it.
Does anyone know of a simple proof of this fact?

I don't know,the proof using Euler-Lagrange equation is definitely a solid one and is quite general,since it makes use of the definiton of a length element in a plane plus the variational principle imposed to the legth of a plane curve:
This method is standard for solving such geometry problems,think of the brahistochrone problem,of the Fermat principle,how could you do it else??
The fact that the shortest distance between two points is a straight line (segment whose ends are the 2 points) can be proven geometrically quite simple.Think of two fixed points and u wann go from one to another on the shortest path possible.Chose the straight line and two joint segments which have the opposite ends as the two points.U have a triangle and use the triangle's inequality to find that the shortest distance is definitely the segment which unites the 2 points,as it is one side of a triangle and the other possibility would imply 2 sides wnd would be more (in length) than one side.
And you can think of generalizing this constructive method for any (continuous/smooth) curve uniting the 2 points.It's just building a number of triangles and apply the triangle's inequality.
And i hope u know how to prove that the sum of 2 triangle's sides are larger and at minimum equal to the other side.Generalized Pythagora's theorem?? :wink:

Daniel.
 

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