Given 3 points A (lat1, lon1), B(lat2, lon2), O(lat3,lon3) on earth with geometric location longitude and latitude and a distance d, where O is middle point of A and B. Let GCD denote the great circle distance between 2 points on earth. Find point P,Q such that $$GCD(P,A) = GCD(P,B), GCD(P,O) = d,$$ and $$GCD(Q,A) = GCD(Q,B), GCD(Q,O) = d.$$
Figure: $\qquad \qquad$
It is straightforward if the question is asked on plane. But I have no idea to find 2 points P and Q on sphere. Is there formula for P and Q in general? Thank you in advance!