Do you know the definition of a 'great circle'?
On the sphere
. . there are many great circles that intersect at
Find coordinates for the other points where these circles all intersect.
i am not sure I understand this problem.
Consider the planet Earth.
There are many circles of longitude (which are all great circles)
. . from the Prime Meridian to the International Date Line.
They all intersect at the North Pole and the South Pole.
We are given a sphere.
The point (3, 4, -12) is a "pole" of this globe.
The other pole is diametrically opposite.
Where is it?