Assuming A is unique it must be on the line through OP. (Otherwise by symmetry if A were on one side of the line you could reflect any A,B,C over the line.)
The point A appears to be at a distance such that p/r = r/|AO|
This ratio also appears to be constant.
I haven't tried to find if A is unique, or if there could be positions off the line OP.
For a given P and R and any position of A, the maximum of (b+c)/a seems to occurs when BC is perpendicular to OP and the minimum when B and C are on OP. The minimum appears to be the constant p/r.
Posted by Jer
on 2015-03-21 20:45:12