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Re-deriving Niele Approximation with Law of Tangents

I am working through Niele's Rule regarding an approximation for elliptical motion (assume constant angular motion around the empty focus of the ellipse and use Niele's Rule to compute the approximate true anomaly).

A screenshot of the relevant quantities is attached (with a couple edits). This image comes from a book page at the following link: http://librarum.org/book/10914/16

(Although Niele's Rule is not a true solution for elliptical motion per equal-area-in-equal-time, I am still trying to work through the derivation.)

The author states that this relationship is a consequence of the Law of Tangents. Sounds simple enough, but I am having a heck of a time trying to confirm it.

Referring to Figure 2.1, to use the Law of Tangents, we need an angle *internal* to the triangle, so define angle J as the supplemental angle to $\displaystyle \theta$. Also define r1 and r2.

The Law of Tangents can then be applied:

$\displaystyle \frac{r1 - r2}{r1 + r2} = \frac{tan[\frac{1}{2}(M - J)]}{tan[\frac{1}{2}(M + J)]}$

The distance between the two foci of the ellipse is 2ae, and r1 + r2 = 2a .

And J is $\displaystyle \pi -\theta$. So the equation can be re-written as follows:

$\displaystyle \frac{r1 - (2a-r1)}{r1 + (2a-r1)} = \frac{tan[\frac{1}{2}(M - (\pi - \theta ))]}{tan[\frac{1}{2}(M + (\pi -\theta))]}$

$\displaystyle \frac{2r1 - 2a}{2a} = \frac{tan[\frac{1}{2}(M - \pi + \theta)]}{tan[\frac{1}{2}(M + \pi -\theta)]}$

1) If I expand the $\displaystyle M - \pi -\theta$ terms first, and then apply the tangent half-angle identity, this equation gets pretty ugly.

2) If I apply the tangent half-angle identity first, and then work through the $\displaystyle M - \pi -\theta$ terms, the equation gets pretty ugly.

I am wondering if I have taken a detour and missed a small simplifying assumption that will yield the desired equation. I seem to be moving further *away* from the desired solution.

Any suggestions?