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?
A polygon is called convex if every interior angle is less than \(180^\circ\), i.e: the shape doesn’t “bulge inwards". Show that if a quadrilateral \(ABCD\) has a convex quadrilateral \(EFGH\) situated completely inside it, then the perimeter of \(ABCD\) is greater than the perimeter of \(EFGH\). You might want to remind yourself of the triangle inequality: in any triangle \(DEF\), the side \(DE\) is always shorter than going around the other two sides, so \(DE < DF + FE\).
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?