As a sequel to Zeta-Flux's triangle puzzle I present a similar problem which is important for networks. It was given by no less our Mentor, Pierre de Fermat himself, to one of Galileo's pupils (name witheld)

Problem:

What is the shortest road network that will join three cities situated at the vertices of a triangle.

In other words, geometrically, it boils down to getting the minimum sum of distances from a point within, to the vertices of a given triangle and to find and construct the point .

Hint: The geometrical solution to find the point is by far the simplest

Mally