Yaglom's road

Northville and Southville lie on opposite sides of the Sticks River. The townsfolk want to build a road and a bridge to facilitate travel between their towns while minimizing the distance along the road. [You can place the towns where you like by dragging the GREEN dots. You can move the bridge by dragging the BLUE dot.] Where should the bridge be located? What is true about the solution? Can you prove your assertion?