I'd greatly appreciate any help with the following problem:
Let be a line and a point not on the line.
Prove that there is no such point such that
In other words, prove that there is no point on the line, such that the distance from to is greater than the distances from all the other points of the line to the point .
Just to illustrate the problem, we may take , and in fact .
However, that is not always true for , for there is a point such that :
I have tried to prove this problem by supposing that the opposite is true and then arriving at contradiction, and tried to use the triangle inequality somewhere on the way, but so far none of my attempts have been successful.
Thanks a lot!