Once in a math interview in college my friend was asked the question,
"If there are $100$ points that are chosen on a plane randomly such that no three points are collinear. Can we always draw line that divides those $100$ points into $50$ points on either side of the line ?"
He answered yes in return and interviewers proceeded to ask him other questions.
Well intuitively I feel there must always be line that can divide $2n$ points into half, that is if no three of the points are collinear. Is there any way to prove this? I am stuck on what the approach of the proof should be, also I can't understand whether a proof is necessary or not? Any ideas, corrections or suggestions are greatly appreciated.
Thanks in advance!
Take a line that isn't parallel to the lines formed by the points. Then slide it until it divides into the two desired subgroups. At each sliding step, only one point can be passed. In fact, if you overpass more than one in a single step, you can do an intermediate one or the two point lie on the line at the same time (this contradicts the fact that the line is not parallel to the lines formed by the points).