Shortest distance between two points. Line?

[Suicidal]

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?

 

[dextercioby]

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??

Daniel.

 

[M98Ranger]

There are a few different ways to prove that the shortest distance between two points is a line. One of the simplest ways is to use the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Now, imagine that we have two points, A and B, and we want to find the shortest distance between them. We can connect these two points with a line and create a right triangle. The length of this hypotenuse is the shortest distance between A and B.

Next, let's label the length of the hypotenuse as d, and the lengths of the other two sides as x and y. We can then apply the Pythagorean theorem to this triangle:

d^2 = x^2 + y^2

Since we want to minimize the distance d, we can treat x and y as variables and use the method of calculus to find the minimum value of d. Taking the derivative of both sides with respect to x, we get:

2d * dd/dx = 2x

dd/dx = x/d

Similarly, taking the derivative with respect to y, we get:

dd/dy = y/d

To find the minimum value of d, we need to set both derivatives equal to 0. This gives us the following equations:

x/d = 0
y/d = 0

From these equations, we can see that the only solution is when x = 0 and y = 0, meaning that the minimum value of d occurs when x and y are both 0. In other words, the shortest distance between points A and B occurs when the line connecting them is a straight line.

This proof may seem complex, but it is based on the fundamental principles of geometry and calculus. I hope this helps clarify the concept for you.