Where Do You Put the Depot to Serve Three Towns? Not Where You Think
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
Finding the location whose total distance to three fixed points is smallest has a surprising answer, a beautiful construction and a physical model made of string and weights.
The question
Given three points, find the single point whose distances to all three, added together, are as small as possible. The obvious guesses are wrong. The centre of the triangle formed by the three, meaning the point where the lines from each corner to the opposite midpoint meet, is not it. Neither is the centre of the circle passing through all three. The correct location has a property that is easy to state and not at all obvious in advance, which is that the three lines from it to the given points meet at equal angles of one hundred and twenty degrees.
The exception and the construction
Two facts complete the answer:
- •If any angle of the triangle is one hundred and twenty degrees or more
- •Then the best location is at that corner itself
- •Otherwise it lies strictly inside the triangle
- •Build an equilateral triangle outward on each side
- •Join each new outer corner to the opposite original point
- •Those three lines cross at the point sought
The soap film model
A physical demonstration makes the answer intuitive and was used long before anybody could compute it. Drill three holes in a flat board at the positions of the points, thread a string through each, tie all three together above the board and hang an equal weight from each string below. The knot settles at the position that minimises the total length of string above the board, which is exactly the point in question, because the system settles where potential energy is lowest. Dipping a frame in soap solution produces the same answer, since a film minimises its own area.
Why the obvious guess fails
Understanding why the centre of the triangle is wrong makes the real answer less surprising. The centre where the lines to the opposite midpoints meet minimises the sum of the squares of the distances rather than the sum of the distances themselves, which is a different quantity and a different point. Squaring penalises a single long distance far more heavily, so that point is pulled towards whichever corner is furthest away. Minimising plain distance treats every unit equally, so the answer sits closer to wherever two of the points are clustered, which is what the equal angle condition produces.
Why it matters beyond three points
The question generalises into a genuinely hard problem with enormous practical importance. Asking for the shortest network connecting any number of fixed points, allowing extra junctions to be added wherever they help, is a famous problem in optimisation, and every added junction obeys the same one hundred and twenty degree rule. The general version is computationally difficult, meaning no efficient method is known for large numbers of points, and it underlies the design of pipeline networks, circuit layouts, road connections and phylogenetic trees. The three point case is where the whole subject starts.
The takeaway
The point minimising total distance to three others sits where the three lines to them meet at one hundred and twenty degrees, unless a triangle angle reaches that size, in which case the answer is that corner. Building outward equilateral triangles and joining across locates it. Hanging equal weights on strings through three holes finds it physically, and the general version underlies network design.