Three distinct segments are chosen at random among the segments whose endpoints are the vertices of a regular $10$-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?

1. In a regular $10$-gon inscribed in a unit circle, the segment joining vertices $k$ steps apart has length $\ell_k=2\sin(18^\circ k)$: $\ell_1\approx0.618$, $\ell_2\approx1.176$, $\ell_3\approx1.618$, $\ell_4\approx1.902$, $\ell_5=2$. There are $10$ segments of each type $k=1,\dots,4$ and $5$ of type $5$, so $\binom{45}3=14190$ choices.
2. Three segments fail to form a triangle exactly for the types $(1,1,3),(1,1,4),(1,1,5),(1,2,4),(1,2,5)$, with $\binom{10}2\cdot10=450$, $450$, $\binom{10}2\cdot5=225$, $10^3=1000$, $10\cdot10\cdot5=500$ triples. The total is $2625$.
3. The probability is $1-\dfrac{2625}{14190}=\boxed{\dfrac{771}{946}}$.