Area distribution of inscribed triangles in a circumference

[Alex_Shev]

TL;DR Summary

Distribution of inscribed in a circumference triangles by areas

Three randomly selected points on a circumference form an inscribed triangle with some area.
What is the probability that the resulting triangle will have an area greater than the predetermined value?

Many attempts has been made by me to solve this problem but no success at all. By solution I mean only obtaining an analytical dependence in general form.

Is there any way to solve it using only a pencil and a sheet of paper and no computer?

Any suggestions are welcome or maybe you already know how to solve it.

 

[renormalize]

Many attempts has been made by me to solve this problem but no success at all. By solution I mean only obtaining an analytical dependence in general form.

Please show us your best attempt to solve this, using LaTeX formulas if at all possible.

 

[Alex_Shev]

Here is my best shot. Trying to demystify the Bertrand paradox I solved the following problem. What is the probability that two randomly chosen dots on a circumference form a chord with the length greater then the predetermined value? Having this solution I got, without any effort, the сhord length distribution function and then the density of distribution of chords along lengths.

Now I knew all about chords and I decided to use this knowledge to solve the problem in question. Two dots form a chord which is also one side of a triangle and the rest we have to know – where the third dot can pop up to give us the desired triangle. But I ran into the equation of the fourth degree which cannot be solved in general form!

 

[renormalize]

But I ran into the equation of the fourth degree which cannot be solved in general form!

You're describing your math with words. Can you display the actual equations?

 

[Alex_Shev]

OK. I don't need advice on solving fourth degree equations.
Can YOU solve the problem or can not? That's all I want to know. I cant.
Can you try to solve it, just in case, maybe it's as simple as 2x2=?
By the way, my hypothesis is that this problem cannot be solved with a pencil and a sheet of paper!
And one more thing. I have formula for the density of distribution of triangles by area. But I don't like the way I got it.

 

[renormalize]

Can YOU solve the problem or can not? That's all I want to know. I cant.

If you can't at least expend some effort to, for example, post your fourth-degree equation and it's derivation, why should anyone on PF go the trouble of offering you constructive feedback?

 

[Alex_Shev]

I don't want any effort! I simply want to know can it be solved with a pencil and a sheet of paper. That's it.
With the help of computer I managed to construct a formula for the density of distribution of triangles by area.
it looks perfect, not a single nameless coefficient, just the area of a circle and the maximum area of an inscribed triangle.

[Insult deleted by the Mentors]

 

[renormalize]

With the help of computer I managed to construct a formula for the density of distribution of triangles by area.

But if you don't explicitly post that formula how can anyone possibly know if it's derivable "using only a pencil and a sheet of paper and no computer"?

 

[Vanadium 50]

I agree with @renormalize that solving an invisible equation is tough.

As always, we can rely on the wisdom of Barbara Billingsley.

 

[Alex_Shev]

But if you don't explicitly post that formula how can anyone possibly know if it's derivable "using only a pencil and a sheet of paper and no computer"?

That’s what I’d like to know – will it be the same formula or not.

I thought that maybe someone would be interested in the problem and would try to solve it just for themselves, not for me, and then we compare our results.

It’s like with the Bertrand paradox. I decided to solve it myself, and did it. I got the solution different from a bunch of others, and all the mistakes that were made by others immediately became clear. But if I took any of known solutions – never got I many interesting things.

 

[renormalize]

I thought that maybe someone would be interested in the problem and would try to solve it just for themselves, not for me, and then we compare our results.

Here is my understanding of the question you are asking:
Three points $p,q,r$ are placed at random on the perimeter of the unit circle (radius $1$ and area $\pi$). These points define the vertices of a random triangle that encloses an area $A(p,q,r)$. Now choose any specific area-value $A_0$ in the range $0<A_0<\pi$. You desire to find the probability $P(p,q,r,A_0)$ that $A(p,q,r)>A_0$.
If this is correct, in order to check your work and offer constructive feedback, I ask that you to display here:

1. Your definition of the random distribution from which the three points $p,q,r$ are chosen.
2. Your explicit formula for the triangle area $A(p,q,r)$.
3. Your explicit formula for the probability $P(p,q,r,A_0)$ that you want compared/verified.

You said "Many attempts has been made by me to solve this problem but no success at all." I infer that you've looked at this for some length of time, but I'm just starting out. It therefore seems only fair to ask that you go first and share your formula for the probability.

 

[Alex_Shev]

1. Sorry, but I'm too old to start studying Latex
2. I'm not a mathematician at all
3. I hate The Probability Theory (no, no, no ... you are wrong, another reason)
4. 0< Ao <1.299 because inscribed triangle has this maximum area if R = 1. The rest is OK.
5. Points obey the uniform distribution law from 0 to 2πR or from 0 to 2π if it's angles
6. As I said I can't solve the problem so I have no formula for the probability P(p,q,r,Ao)
7. It's very simple to obtain a formula for the area of a triangle in terms of three angles, but it is not possible to express the dependence Ao(A,p,q,r) from it.
8. The probability in question interests me because it is easy to obtain from it the distribution function and then the distribution density. The distribution density is my actual interest!
9. We can compare then the construction one (obtained God knows how) and the actual one, obtained by some mathematician.

 

[fresh_42]

...
3. I hate The Probability Theory (no, no, no ... you are wrong, another reason)
...

Whatever the reason is, this forum is obviously not suitable for you, since it is all about probability theory.

Our rules require something to discuss about. This can be a scientific paper, a question from a curious kid, or calculations that somebody has done and is stuck at or wants us to confirm it.

Now, a) you didn't provide any scientific sources, e.g. about probability distributions for randomly choosing points on a circle, b) you are too old to be a curious kid, c) you refuse to show us your calculations although you have been asked several times, and you got angry - even on Russian - instead of contributing to your own question.

There were so many rule violations that it doesn't make sense to circle around any longer. We are a forum that is based on dialogues, we do not solve other people's problems. My suggestion is to post your question on MSE. They often like to brag about their knowledge and solve it for you. However, they also require some effort from your side, so it could be a matter of time or luck whether a moderator there catches you before some of the show-offs there answers your question. We here believe in dialogues so that readers can follow the thoughts and steps. And one last hint: MSE doesn't accept Russian rants either, and I'm afraid they use LaTeX, too.

Since we don't get any further, I closed this thread for now.