In national mathemathics magazine i found some cool task. On a plane we have finite amount of points. Any three of them are not colinear. Show that there exist circle ( formed with three or more points ) that doesn't contain other points in it.
My logic is quite simple. Assume we have some circle created from 3 or more points. If in this circle(interior) exist some point, we take any two points from the border and one from the inside and create new circle.
And we can repeat it until we have circle without inner points.
Is there more elegant way to prove such things?