I just answered this question in chat, so I thought it would be worth posting the answer here.
The horizontal position of a vertical chord of length $c$ in a circle of radius $r$ would be $\pm\sqrt{r^2-\frac{c^2}4}$.
Thus, the distance, $d$, between chords of length $c_1$ and $c_2$ (with $c_1\ge c_2$) would be
$$
d=\sqrt{r^2-\tfrac{c_2^2}4}\pm\sqrt{r^2-\tfrac{c_1^2}4}\tag1
$$
The "$\pm$" depends on whether or not the chords appear on the same side of the center of the circle. In either case, we have
$$
\frac{c_1^2-c_2^2}{4d}=\sqrt{r^2-\tfrac{c_2^2}4}\mp\sqrt{r^2-\tfrac{c_1^2}4}\tag2
$$
To verify $(2)$, simply multiply $(1)$ and $(2)$.
Average $(1)$ and $(2)$ to get
$$
\frac{c_1^2-c_2^2+4d^2}{8d}=\sqrt{r^2-\tfrac{c_2^2}4}\tag3
$$
Thus,
$$
\begin{align}
r^2
&=\frac{\left(c_1^2-c_2^2+4d^2\right)^2+16c_2^2d^2}{64d^2}\tag{4a}\\
&=\frac{c_1^4+c_2^4+16d^4-2c_1^2c_2^2+8c_1^2d^2+8c_2^2d^2}{64d^2}\tag{4b}\\
&=\frac{\left(c_1^2+c_2^2\right)^2+8\left(c_2^2+c_1^2\right)d^2+16d^4-4c_1^2c_2^2}{64d^2}\tag{4c}\\
&=\frac{\left(c_1^2+c_2^2+4d^2\right)^2-4c_1^2c_2^2}{64d^2}\tag{4d}\\
&=\frac{\left((c_1+c_2)^2+4d^2\right)\left((c_1-c_2)^2+4d^2\right)}{64d^2}\tag{4e}
\end{align}
$$
Explanation:
$\text{(4a):}$ solve $(3)$ for $r^2$
$\text{(4b):}$ expand
$\text{(4c):}$ collect
$\text{(4d):}$ collect
$\text{(4e):}$ apply $a^2-b^2=(a+b)(a-b)$
Therefore,
$$
r=\frac{\sqrt{\left((c_1+c_2)^2+4d^2\right)\left((c_1-c_2)^2+4d^2\right)}}{8d}\tag5
$$
In $(5)$, it doesn't matter whether $c_1\ge c_2$ or not.
