Old Smokey,
You misunderstand the point I was trying to make.
As I said the abeam point will be the point on the surface of the earth where the great circle from A to C intersects a great circle passing through B at an angle of 90 degrees. There are only two possible places on the earth where that happens and generally one will be a lot further from A than the other!!.
I was responding in simple terms to the point that there could be a GC from A to the abeam point and then a second different great circle from the abeam point to C. If this were the case then the abeam point could be in a number of places at various distances from A, B and C but each of the unlimited possible points satisfies the "Abeam B" requirement. Therefore that is not a good idea.
Simple non-trig tables answer.
PS You are correct with the chart however if you want to be pedantic, man started navigating by great circle from the time that his fellow man was able to point the direction out i.e Over there = great circle. It was only the invention of the compass and lat/long that complicated matters.
