In Hyacinthos message #9139, you wrote:
>> Let AX and BY are the bisectrix of triangle ABC,
>> O, R - the center and radius of its circumcircle,
>> I1, r1 - the center and radius of excircle
>> touching the sideline AB, A', B', C' - the
>> touching points of excircle with BC, CA, AB.
>> Then next conditions are equivalent:
>> 1. R=r1.
>> 2. O is in line XY.
>> 3. O is the orthocenter of A'B'C' (I suppose
>> also that A'B'C' is autopolar with respect to
>> circumcicle of ABC).
>> 4. cosA+cosB=cosC.
You can add the following equivalent condition:
5. If I is the incenter of triangle ABC, and Hb
and Hc are the feet of the altitudes from B and C,
then I lies on HbHc.
The equivalence of the conditions 2. and 5. was the
subject of Problem 4 in the Bundeswettbewerb
Mathematik (German National Mathematics Olympiad)
2002, 2 round. In fact, it was a very hard problem;
I was lucky enough to know the method of trilinear
coordinates to solve it!