Let Γ be a circle with center O and radius r. Let P be a point inside Γ
( different from O ) with |OP| = p.

a) Prove there exists a point A outside Γ such that for all chords BC
of Γ through P the quantity (b+c)/a is constant ( where a, b, and c
are the side lengths of ΔABC ).

b) What is the constant in terms of p and r?

c) Prove that the point A is unique.
  

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