(In reply to
Possible Solution by Harry)
Liked your construction of the circle through points B, C, and O to find point A and the value of the constant (b+c)/a for that point A.
The first thing I thought of when I saw the problem was "point P inside Gamma and point A outside Gamma" and that P and A should be inverses with respect to Gamma.
You lost me about half way through the uniqueness part.
I used Geometer's Sketchpad to construct an ellipse.
1) Constructed a fixed length segment B'C'.
2) Constructed point A' such that triangles ABC and A'B'C'
are directly similar.
3) A' then traces out the ellipse as B traces out Gamma.
Edited on March 25, 2015, 10:23 pm

Posted by Bractals
on 20150325 22:18:18 