Show that for any three points on the plane \(A,B\) and \(C\), \(AB \ge |BC - AC|\).
Two villages lie on the opposite sides of a river whose banks are straight lines. A bridge is to be built over the river perpendicular to the banks. Where should the bridge be built so that the path from one village to the other is as short as possible?
Quadrilateral \(ABCD\) is situated completely inside a quadrilateral \(EFGH\). Prove that the perimeter of \(ABCD\) is smaller than the perimeter of \(EFGH\).
There are \(n\) mines and \(n\) cities scattered across the land, it is known that no three objects (mines, or cities) belong to one line. Every mine has to have a rail connection to exactly one city. Railways have to be straight and cannot cross other railways. Is it always possible?