Part 1: Does the arithmetic mean of sides right triangles to the mean of their hypotenuse converge?

Quantities $x=a/c$ and $y=b/c$ are the legs of a pythagorean triangle having unit hypotenuse, hence they are the coordinates of points lying on a unit circle centred at the origin, and $x=\cos\theta$, $y=\sin\theta$ with $\pi/4<\theta<\pi/2$.

It is reasonable to think that, at least in the case of primitive triples, those points are spread evenly on that arc. In that case their average values are: $$ \langle x\rangle={\int_{\pi/4}^{\pi/2}\cos\theta\,d\theta\over\int_{\pi/4}^{\pi/2}d\theta}= {4-2\sqrt2\over\pi}\approx 0.372923, $$ $$ \langle y\rangle={\int_{\pi/4}^{\pi/2}\sin\theta\,d\theta\over\int_{\pi/4}^{\pi/2}d\theta}={2\sqrt2\over\pi}\approx 0.900316. $$ One should then justify that $\langle a\rangle/\langle c\rangle$ and $\langle a/c\rangle$ have the same limiting value, but that also seems very reasonable. I ran a simulation up to $k\approx800000$ and found the encouraging results: $$ s_k\approx0.373,\quad l_k\approx0.900. $$