20070310, 08:29  #1 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Network problem
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 
20070310, 10:44  #2 
"Lucan"
Dec 2006
England
14512_{8} Posts 
120 degrees between the arms gives a local minimum

20070310, 11:03  #3 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
unless the triangle has an angle >120 degrees,
in which case the network consists of the two shortest sides Will you believe I didn't Google this please? 
20070310, 14:42  #4 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
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 
20070310, 14:59  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Half Right.
Quote:
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 Last fiddled with by mfgoode on 20070310 at 15:01 Reason: Add on 

20070310, 17:15  #6  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
edges of the triangle. The circumcircles of these triangles intersect at the desired point Last fiddled with by davieddy on 20070310 at 17:43 

20070310, 17:41  #7  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
the network consists of the two shortest sides Last fiddled with by davieddy on 20070310 at 17:41 

20070311, 18:55  #8 
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
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?

20070311, 20:58  #9 
"Lucan"
Dec 2006
England
14512_{8} Posts 
a>b

20070312, 17:58  #10 
Feb 2007
2^{4}×3^{3} Posts 

20070312, 20:12  #11  
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Quote:
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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