$4n$ points are uniformly distributed on a circle. Parabolas are drawn in the manner shown below with example $n=4$.
The parabolas' vertices are at the center of the circle. The parabolas have a common line of symmetry. Each parabola passes through two of the designated points on the circle. (Two parabolas are degenerate and look like lines.)
Call the lengths of the curves $l_1, l_2, l_3, ..., l_{4n}$.
What is the exact value of the radius $r$, such that $L=\lim\limits_{n\to\infty}\prod\limits_{k=1}^{4n} l_k=1$ ?
By experimenting on desmos, it seems that $r\approx0.975399$.
(I believe the number of points does not have to be a multiple of $4$. If it is any even number, then with the above value of $r$, the product of the lengths approaches $1$. But, for the sake of simplicity, I am just asking about the case with $4n$ points.)
My attempt
Let $\theta=\dfrac{k\pi}{2n}$. We have
$L=\lim\limits_{n\to\infty}r^{4n}\left(\prod\limits_{k=1}^{n-1}\int_0^{\cos\theta}\sqrt{1+\left(\dfrac{2\sin\theta}{\cos^2\theta}x\right)^2}dx\right)^4$
With some help from Wolfram, and simplifying by taking the 4th root:
$L=\lim\limits_{n\to\infty}r^{n}\prod\limits_{k=1}^{n-1}\left(\dfrac{1}{2}\sqrt{1+3\sin^2{\theta}}+\dfrac{\cos\theta}{4\tan\theta}\sinh^{-1}{(2\tan\theta)}\right)=1$
I do not know how to evaluate the product, and I do not know how to find the exact value of $r$.
Context: This is part of a series of investigations I have been doing (just for fun) about infinite products of areas and lengths in a circle. Here are other questions: question1, question2, question3, question4, question5, question6.
EDIT: The previous version of this question was about parabolas in a semicircle. I mistakenly believed that, if the radius is $0.975399...$ then the product of the lengths approaches $1$. Now I believe that (for the semicircle) the product of the lengths approaches the value of the radius.
