Ten days have passed (as of a few hours hence) with no answer to this one on stackexchange. My question at the very bottom will be: What particular commonality between these two problems causes this same bit of algebra to occur in both places?
The Villarceau circles are things whose existence is surprising. To find radii of Villarceau circles, I stupidly went through a bit of trigonometry and got a much simpler result than I expected, and then realized there was a glaringly obvious way to do it that I hadn't thought of.
In the $xyz$-space imagine a circle of radius $r>0$ in the $xz$-plane, whose center is at a distance $R>r$ from the $z$-axis, and revolve it about the $z$-axis, getting a torus embedded in $\mathbb R^3$. The intersection of that surface with the $xz$-plane is two circles not crossing each other. A line $\ell_1$ touches one of those circles on one side and another on the other side, and that line is in a plane parallel to the $y$-axis, and the intersection of that plane with the torus is the union of two Villarceau circles. So I thought: let's draw a line $\ell_2$ touching both circles on the same side, and the other line $\ell_3$ touching both circles on the same side, and the distance from the intersection of $\ell_1$ with $\ell_2$ to the intersection of $\ell_1$ with $\ell_3$ is the diameter of the Villarceau circle. So I thought: first, the distance from the center to the point of tangency of $\ell_1$ with one circle, is $$\sqrt{R^2-r^2}.\tag{1}$$ Add to that the distance the distance from that point of tangency to the point of intersection of $\ell_1$ with the nearest of those two parallel $\ell$s, and that distance is $$ \frac{r^2}{R+\sqrt{R^2 - r^2}}.\tag{2} $$ So the sum of $(1)$ and $(2)$ involves finding a common denominator, doing some routine cancelations, getting $$ \frac{R\left(R+\sqrt{R^2-r^2}\,\right)}{R+\sqrt{R^2-r^2}} $$ and one more cancellation gives you $R$. Then I realized that the obvious way to see that the radius is $R$ doesn't involve doing any of that.
But I recognized that bit of trivial algebra from a routine calculus problem: $$ \frac{d}{dx} \log\left(x+\sqrt{x^2-1}\,\right). $$ Going through the usual algorithms, you get this down to $$ \frac{x+\sqrt{x^2-1}}{\sqrt{x^2-1}\left(x+\sqrt{x^2-1}\,\right)} $$ and again to one last cancelation to get $\dfrac{1}{\sqrt{x^2-1}}$.
SO MY QUESTION IS: What particular commonality between these two problems causes this same bit of algebra to occur in both places?
