Characterise all pairs of n/m stars that have the same inner radius Geometry, algebra, and examples
Let n and m be integers, with 2 ≤ m < n/2. Consider the bounding polygon of an n/m star (that is, a star with n points each of which connects to the two points ±m away) inscribed in the unit circle. Such a bounding polygon has 2n points, n on the unit circle and n on an inner circle with:
    $Inner Radius = \frac{\cos{\left(\frac{\pi m}{n}\right)}}{\cos{\left(\frac{\pi (m-1)}{n}\right)}}$  
    
    
    
E.g., inner radius of 5/2 star is ½(3–√5) ≈ 0.381966, that of 8/2 star is √(2–√2) ≈ 0.765367, and that of 8/3 star is √(1–√½) ≈ 0.541196. 
I wish to characterise the different stars that have the same inner radius. Two series of pairs are known: ∀ integer i ≥ 2, stars (6i–2)/i and (18i–6)/(6i–2) have the same inner radius, as do (6i–4)/i and (18i–12)/(6i–3). Proof that these pairs do match is boringly elementary, given the identity Cos[θ] Cos[φ] = ½Cos[θ+φ] + ½Cos[θ–φ], and at jdawiseman.com.
But are there any other non-trivial matches, perhaps another series, perhaps sporadic? 
(I know that any other matches must have inner radius > 0.998122, and of course strictly < 1. So any other matches shown as a graphic smaller than about 2k pixels across must look like a circle — practical rather than proper progress.)
The Question

Characterise all pairs of stars, n₁/m₁ and n₂/m₂ (all integer), such that the stars have the same inner radius. It is known that there are two series of such pairs of stars, {(6i–2)/i, (18i–6)/(6i–2)} and {(6i–4)/i, (18i–12)/(6i–3)}. Are there any other series? Are there any sporadic matches?

 A: [Expanding some on my comment of a few days ago]
This is a special case of a problem that was solved by Gerrit Bol in 1936 [B];
that nearly-forgotten result was rediscovered, using slightly different
methods, by Bjorn Poonen and Michael Rubinstein [PR].  (As it happens
I used such coincidences in my own work a few years ago [E].)
They find all ways that three diagonals of a regular polygon can meet 
at a point: there are several infinite families (comprising algebraically "trivial" 
solutions that are not always geometrically obvious, plus the four "nontrivial"
families of Table 3 on page 12), and 65 sporadic solutions (Table 4 on page 13).
If two stars have the same inner and outer radii then we can rotate them
so they share an inner vertex; then the outer vertices are contained
among the vertices of a regular polygon, and the shared inner vertex
is on at least four diagonals $-$ indeed at least five if we include
the line of symmetry (and double the order of the regular polygon
if necessary).  Four infinite families (all symmetrical) of such
quintuple intersections are listed in Table 6 (page 16), and a
finite computation limits further sporadic solutions to denominators
18, 24, and 30 (pages 15-16).  If you've already computed far enough
to find any sporadic solutions then the infinite families must account
for everything else.
References
[B] Gerrit Bol: Beantwoording van prijsvraag no. 17, 
Nieuw Archief voor Wiskunde 18 (1936), 14-66.
[PR] Bjorn Poonen and Michael Rubinstein:
The Number of Intersection Points Made by the Diagonals of a Regular Polygon,
SIAM J. Discrete Math 11 (1998), 135-156
(http://www-math.mit.edu/~poonen/papers/ngon.pdf).
[E] Noam D. Elkies: 
On some points-and-lines problems and configurations, 
Periodica Mathematica Hungarica 53 #1-2 (2006), 133-148
(arXiv:MG/0612749).
