I
agree with Daniel & Charlie that triangle MNR is isosceles:

ratio = (NM + MR)/NR = (sin/R + sin/N)/sin/M

= (sin/R + sin/N)/(sin(/R
+ /N)

This symmetric function in /R and /N will be maximum

when /R = /N, giving NM = MR, so that R lies on the circle, C,

in X, with centre at M and radius MR.

Using the x-y plane as X, with the origin, O, at the projection
of N on to this plane, let N have coordinates (0, 0, n) and let
M be the point (m, 0, 0). Then R lies on the circle with centre
M and radius r = sqrt(n^2 + m^2).

Since NM and therefore MR are fixed lengths, the ratio
(NM + MR)/NR will be maximum when NR is minimum, i.e.
when R lies on the line MO produced, at which point

OR = r – MO = r – m where r = sqrt(n^{2}
+ m^{2})

and the ratio is then equal to 2r/sqrt(n^{2}
+ (r – m)^{2})

which simplifies to:ratio =
sqrt(2/(1 – m/r))

With Charlie’s values of n = 3 and m = 5, these formulae