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Puzzles Minimum Curve Bisector 
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What is the curve of minimum length which bisects the area of an equilateral triangle? | |
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Comments | terveloc |  18:05:00, 13 Feb 08 |  Newbie
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| Now I guess I need to optimize for the best choice of a
...though I kind of feel like I'm killing an ant with an A-bomb. There has to be an easier way to do this.
| | | | terveloc | 18:03:04, 13 Feb 08 | Newbie
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| Here's what I have...
Assume the length of each triangle side is x.
If you start at the point a distance a<x along the rightmost edge from a vertex, then the shortest curve to bisect the triangle is defined as a polar function r with origin at the mentioned vertex.
r = a*sin(theta)-(1/3)*sqrt(3)*cos(theta)*a+(1/2)*cos(theta)*x^2
I used calculus of variations. I know this looks ugly, but this is what I've got. Does this look at all right?
| | | | terveloc | 16:31:15, 13 Feb 08 | Newbie
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| To Singer: My first thought in this kind of optimization problem is calculus of variations, but the geometric constraints are a little tricky. So, first, perhaps determine what half of the area *is*...then. parameterize two sides of the triangle from a common vertex. Then find the family of curves that cut a wedge (of sorts) with the desired area. Then, optimize with calculus of variations. The only issue is that I am not certain such a method would be able to pinpoint a unique* solution.
*Equivalent by symmetry.
Any thoughts?
| | | | singer | 06:05:25, 28 Jan 08 |  Newbie
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| this is driving me nuts  I can't think of how to prove it at all- hows it done!??
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