Three points are randomly selected on a circle of radius R. They form an inscribed triangle with some area. What is the probability that the area will be greater (or less) than a given value S. All my attempts to solve this problem have failed. Any idea or maybe a solution?
Square distribution: Three points are randomly selected on a circle of radius R. They form an inscribed triangle with some area.
- Thread starter AlexShev
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Please reply with a clear listing of all of your steps in at least one of your attempts at finding the solution.
Thank you!
Dr.Peterson
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I would take the first point as given (say, at the top), and imagine choosing two angles x and y from 0 to 360 for the locations of the other two points. Then write an equation for the area of the triangle in terms of those two variables.
Show whatever you have tried, and we can offer additional help.
Show whatever you have tried, and we can offer additional help.
Dr.Peterson
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One more question: Where did this come from? If it's your own idea, keep in mind that it's easy make up a problem that can't be solved; if not, I'm guessing it's some sort of contest problem or worse. I've done the part I suggested, but the next step looks more challenging. I'd like to have some reason to think it can be solved (and will be of use), before putting in any more effort.
BigBeachBanana
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The question was posted at another forum.
mathforums.com
Square distribution of triangle
Three points are randomly selected on a circle of radius R. They form an inscribed triangle with some area. What is the probability that the area will be greater (or less) than a given value S. All my attempts to solve this problem have failed. Any idea or maybe a solution?
mathforums.com
Ok. When I run into Bertrand's paradox (circle and inscribed triangle) I decided to try it myself and it turns out that it's very easy to solve. So I have got chord length distribution function and chord length distribution density. Having all these I got curious can I find out something about areas of randomly chosen inscribed triangles. I tried this approach: two points give us a chord and the rest we need to solve the question, simply find out where the third point can pop up to form appropriate triangle. If we have this space on the circle than we have the probability. But I failed.
I almost forget to say. I HAVE the density of distribution of triangles by area and in Excel notation it looks like
f(s)=(1/Smax)*(So-Smax)/So*(s/Smax)^(-Smax/So),
where So - area of disk with radius R, Smax - maximum area of inscribed triangle.
But I don't like the way I got it.
f(s)=(1/Smax)*(So-Smax)/So*(s/Smax)^(-Smax/So),
where So - area of disk with radius R, Smax - maximum area of inscribed triangle.
But I don't like the way I got it.