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 2007-03-10, 08:29 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 205210 Posts Network problem 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
 2007-03-10, 10:44 #2 davieddy     "Lucan" Dec 2006 England 193316 Posts 120 degrees between the arms gives a local minimum
 2007-03-10, 11:03 #3 davieddy     "Lucan" Dec 2006 England 6,451 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?
 2007-03-10, 14:42 #4 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22×33×19 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
2007-03-10, 14:59   #5
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

40048 Posts
Half Right.

Quote:
 Originally Posted by davieddy 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?

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 2007-03-10 at 15:01 Reason: Add on

2007-03-10, 17:15   #6
davieddy

"Lucan"
Dec 2006
England

6,451 Posts

Quote:
 Originally Posted by mfgoode How will you construct the point in relation to the 3 vertices ? Mally
Construct equilateral triangles on the outside of the
edges of the triangle. The circumcircles of these triangles
intersect at the desired point

Last fiddled with by davieddy on 2007-03-10 at 17:43

2007-03-10, 17:41   #7
davieddy

"Lucan"
Dec 2006
England

144638 Posts

Quote:
 Originally Posted by mfgoode 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
When the original triangle has an angle >=120 degrees,
the network consists of the two shortest sides

Last fiddled with by davieddy on 2007-03-10 at 17:41

 2007-03-11, 18:55 #8 davieddy     "Lucan" Dec 2006 England 6,451 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?
 2007-03-11, 20:58 #9 davieddy     "Lucan" Dec 2006 England 6,451 Posts a>b
2007-03-12, 17:58   #10
m_f_h

Feb 2007

6608 Posts

Quote:
 Originally Posted by davieddy What is the shortest road network that will join four cities situated at the vertices of a rectangle of sides a and b?
If I remember well, something looking like >-<
where angles are 120° (?)
I think you can get the result by using a "soap bubble computer"

2007-03-12, 20:12   #11
davieddy

"Lucan"
Dec 2006
England

6,451 Posts

Quote:
 Originally Posted by m_f_h If I remember well, something looking like >-< where angles are 120° (?) I think you can get the result by using a "soap bubble computer"
That's what I guess. Total length a + b*SQR(3)

By twisting the "forks" we get the solution to a typical tetrahedron,
David

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