Prove that the shortest distance between two points is a straight line.?

Answer 1

If you have to go from your location to the antipodal point of the globe, the "shortest (and safest!) path doesn't go through the Earth center! It depends on your "frame", on the definition of "distance" and on the definition of "straight line". But, naturally, if you define the stright line as the path of minimal distance (geodesic), it is a tauthology. For example, the light always follows the path of "minimal time", not of "minimal distance", so you have refraction, diffraction and other phenomenons.

You can also consider the question from another point of view:
If your measure of "distance" to go from A to B in your town is the "time you spend all thing being considered", you have to consider many factors, as for instance:
1)traffic and road works
2) how much you want to pay for code infractions
3) your wife's mood waiting for you...

funny and delicate choices...

Answer 2

Note that any line is an infinite array of points. It may seem that you pull or push a pencil or pen or like along a ruler to draw a straight line. But actually you plot solid points continuously to join them to make a straight line. If there is no such space between any two even in a straight line, then the shortest distance between two points is nothing other than a straight line!

Q.E.D.

Answer 3

If it wasn't a flat plane then it wouldn't be a straight line so the statement would still be true.

Answer 4

It's not true for all cases; only flat planes. The statement is not universally true.