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ganeshie8
 one year ago
Find the curve of minimum length which divides the equilateral triangle into two pieces of equal area (median curve)
ganeshie8
 one year ago
Find the curve of minimum length which divides the equilateral triangle into two pieces of equal area (median curve)

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3dw:1435819109185:dw

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6This must be fairly easy with variational methods but I'm wondering if there's a more elementary method that only uses calculus. I think I can sort of see that at every point the curve must be perpendicular to the line segment connecting it to the vertex, but I wouldn't know how to prove it.dw:1435826489623:dw The image sort of describes my intuition. Since over the small distance r doesn't change, the biggest area we can get is by going perpendicular to r. I guess this would imply that we need to go along a circle centered on the facing vertex.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6"the biggest area we can get is by going perpendicular to r." I meant : the largest amount of area we can get for a given additional length ds.

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.1so i cant make only lne ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3You have a very good imagination @beginnersmind ! For simplicity, I guess we may use the fact that of all the closed curves of given area, a circle has minimum perimeter.

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.1what type of circle ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3@ikram002p median of a triangle does divide the triangle into two parts of equal area, but it need not be of minimum length right

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435827736653:dw

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.1or dw:1435827776826:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3yes i think beginnersmind is referring to the second one

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.1well we always can construct a short cut for curves dw:1435827828432:dw

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6Yeah, that one. I get 0.6734 for length, while using a line takes 0.707 (1 over squareroot 2)

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6"You have a very good imagination @beginnersmind ! For simplicity, I guess we may use the fact that of all the closed curves of given area, a circle has minimum perimeter." Thanks. I think I got it now. Let's assume there's another curve with a shorter length that encloses the same amount of area. Then we could copy it 5 more times to create a closed curve which has the same area as a circle but smaller perimeter, which would be a contradiction.dw:1435828385612:dw

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.1If u wanna the smallest closed curve inside a triangle that touches its edges that would be a circle.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Excellent! that gives us a closed curve inside a hexagon. The closed curve divides the hexagon into two parts of equal area

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6Really cool puzzle. Seems almost impossible at first but actually all you need is a good idea and the fact about the circle having the smallest perimeter for a given enclosed area.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3dw:1435915373490:dw
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