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Let $z_0,z_1,z_2$ be the random points on the circle. We may assume that $z_0=1$, $z_1=e^{i\thh_1}$, $z_2=e^{i\thh_2}$, where $\thh_1,\thh_2$ are iid uniformly distributed on $[0,2\pi]$, and so, by the formula $A_\bigtriangleup=\frac12\,ab\sin C$, the area of the random triangle is
\begin{equation}
	A=2\sin U_1\,\sin U_2\,\sin(U_1+U_2), 
\end{equation}
where the $U_i:=\thh_i/2$'s are iid uniformly distributed on $[0,\pi]$. So, in view of the Euler formula $\sin\thh=(e^{i\thh}-e^{-i\thh})/(2i)$, all the moments 
\begin{equation}
	\E A^n=\frac1{\pi^2}\iint_{[0,\pi]^2}(2\sin u_1\,\sin u_2\,\sin(u_1+u_2))^n\,du_1\,du_2\,
\end{equation}
(with natural $n$) are rational numbers, which are actually diadic, with denominators being natural powers of $2$. This is so because the integrand is a polynomial in $e^{iu_1},e^{iu_2}$ with diadic rational coefficients. 

In particular, for $n=2,4,6$ we have the respective values $3/8,45/126,105/256$ of $\E A^n$.