Why can three paths share a shorter junction?

Connecting three places does not always mean drawing lines directly between pairs. A new meeting point can reduce the total amount of path. This is a geometric idea behind studying economical networks, although real roads have many concerns beyond length.

Imagine three sites at the corners of an equilateral triangle, with each side six length units. Following two sides connects all three and uses twelve units. A third side is unnecessary if the only requirement is that every site can reach every other.

Now place a junction at the triangle's center and connect each corner to it. The three equal branches meet at angles of 120 degrees. Each branch is about 3.46 units, making the total about 10.39 units. More junctions in the picture do not mean more total path.

For this symmetric arrangement, the center gives the least possible total length. Moving the junction toward one corner shortens that branch but lengthens the other two by more altogether. The minimum balances the three directions rather than favoring one.

These conclusions assume straight paths on a flat, empty surface and equal cost per length. Obstacles, slopes or unequal costs change the problem. Even with three sites, a very wide triangle can have its best junction at an existing corner instead of an interior 120-degree meeting point.