Network problem
:smile:
As a sequel to ZetaFlux'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 :coffee: 
[spoiler]120 degrees between the arms gives a local minimum [/spoiler]

[spoiler] unless the triangle has an angle >120 degrees,
in which case the network consists of the two shortest sides [/spoiler] Will you believe I didn't Google this please? 
:wink:
Fair enough I give you the benefit of the doubt as its an old classic problem and hence well known. But that is only half the answer. How will you construct the point in relation to the 3 vertices ? Mally :coffee: 
Half Right.
[QUOTE=davieddy;100431][spoiler] unless the triangle has an angle >120 degrees,
in which case the network consists of the two shortest sides [/spoiler] Will you believe I didn't Google this please?[/QUOTE] :smile: Not quite! Davie you are doing well and keep it up. Lets discuss this further. Which triangle ? You mean the angles contained by the arms at the point. What happens if one of the three angles is equal to 120* and the other two aren't ? Mally :coffee: 
[quote=mfgoode;100444]:wink:
How will you construct the point in relation to the 3 vertices ? Mally :coffee:[/quote] [spoiler] Construct equilateral triangles on the outside of the edges of the triangle. The circumcircles of these triangles intersect at the desired point [/spoiler] 
[quote=mfgoode;100445]:smile:
Which triangle ? You mean the angles contained by the arms at the point. What happens if one of the three angles is equal to 120* and the other two aren't ? Mally :coffee:[/quote] [spoiler]When the original triangle has an angle >=120 degrees, the network consists of the two shortest sides [/spoiler] 
New problem
What is the shortest road network that will join four cities situated at the vertices of a rectangle of sides a and b?

a>b

[quote=davieddy;100545]What is the shortest road network that will join four cities situated at the vertices of a rectangle of sides a and b?[/quote]
If I remember well, something looking like >< where angles are 120° (?) I think you can get the result by using a "soap bubble computer" 
[quote=m_f_h;100614]If I remember well, something looking like ><
where angles are 120° (?) I think you can get the result by using a "soap bubble computer"[/quote] That's what I guess. Total length a + b*SQR(3) By twisting the "forks" we get the solution to a typical tetrahedron, which leads me to doubt Cheesehead's soap film proposal. (See tetrahedral network thread) David 
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