## New answers tagged geometric-probability

1
vote

### The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

Here is an elementary proof using basic trigonometry for the more general formula $p(ab<rc)=\frac{2 \arctan(r)}{\pi}.$
Fixing one point on a circle of radius $r$ and placing the other two at angles ...

6
votes

Accepted

### The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

One can use basic probability theory to prove that
$$P(ab<kc)=\frac{2}{\pi}\arctan k,\qquad k>0.$$
Without loss of generality, the vertices opposite the sides $a,b,c$ are
$$A=e^{2i\beta},\qquad ...

5
votes

### The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

A proof from pure geometry
On a unit circle with centre $O$, draw parallel chords $PQ$ and $P'Q'$ such that $PQ'\perp P'Q$. Chord $MN$ is parallel to $PQ$ and passes through $R$, the intersection of $...

6
votes

The following argument is less or more the same as that of Iosif Pinelis, but with less computations and the symmetry is rather explicit. It may be explained without complex numbers, but the ...

2
votes

Denoting the angles corresponding to edges of side lengths $a,b,c$ with $A,B,C$ respectively, by the sine law:
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2.$$
Thus $c>ab$ amounts to $\sin(...

0
votes

We use the formula (from en.wikipedia.org/wiki/Circumcircle#Other_properties ):
diameter = a * b * c / ( 2 * area )
In our case diameter = 2 and we get:
4 * area = a * b * c
If h is the height of ...

5
votes

Without loss of generality, the three points on the unit circle are $1,e^{is},e^{it}$, where $s$ and $t$ are uniformly and independently distributed on the interval $[0,2\pi]$, so that $a=|e^{is}-1|$, ...

0
votes

### Random partition of an interval – Dirichlet distributed?

Suppose $1\le k_1\le k_2\le \cdots \le k_\ell \le N$ are integers. Then the probability distribution of the vector of lengths of
$$
[0, X_{(k_1)}], [X_{(k_1}, X_{(k_2)}], \ldots, [X_{(k_\ell)}, 1]
$$
...

1
vote

### Random partition of an interval – Dirichlet distributed?

This follows from the properties of the Poisson process. Consider a Poisson process (say, of rate 1), and let its first $N+1$ points be $Y_1<Y_2<\dots<Y_{N+1}=:A$. Then $\xi_0:=Y_1$, $\xi_1:=...

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