G'Day Stallie!
I don't have my books to hand and would rather not derive equations from first principles but...
I can recommend this applet if used with a bit of gumption. The bit you want is called "dads" if I recall correctly and is to better than metre accuracy.
http://www.naco.faa.gov/index.asp?xm...online/compsys
First derive a convenient point on each spherical side using the routine <segment-distance> or take the mid-point. Then using these three calculated points and the three apexes (apices??) find where they intersect using routine <segment-segment>.
I did write an Excel spreadsheet using the Andoyer-Lambert method to do exactly what you ask and I stole the formulae from the Admiralty Manual of Navigation but I'm as far from my computer as you are from a reference library.