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Frank,

As cleverer chaps than me have proved that, all things being equal, the probability of your true position being inside a cocked hat is only around 25%, I wouldn’t be too worried about using fancy maths inside a cocked hat.  I’d stop shooting when I’d taken stars A and B which gives a nice 90 degree cut two position line fix and then get on with the rest of the job.  However, I would MPP that fix with my DR position in the ratio of the quoted 50% accuracies of my celestial and my DR equipment. 

Ignoring my carefully maintained DR position, that gives me a fix 10nm up and 20nm right.  That’s Vulcan 480kts celestial.  For the boys & girls of the Bulldog Breed bobbing up and down on the sea at 4.8kts there might be more time for sophistication. 

If you insist I use three position lines, I’d mark my fix along PL A halfway between where PLs B & C cross it.  That gives me 11nm up and 21nm right. My reason for doing that is the narrower the angle between PLs B&C the accuracy of the BC intersection varies greatly and pulls a centred fix too far from PL A, irrespective of which centre you use.  E.g.  If the angle between PLs B&C approaches zero, they would eventually intersect at infinity, so your fix would be half-way to infinity (joke).  DaveP