Let ”ABC” be a triangle, ”I” its in-centre; 
be the reflections of I in BC, CA, AB respectively. Suppose the circumcircle of triangle 
passes through A. Prove that 
are concyclic, where 
is the incentre of triangle .
Basic Sketch:
Hint 1: I is indeed the circumcenter of 
with circum radius = 2r. 
is the radical axis of the two circles concerned hence the other center has to lie of IA (since IA is perpendicular to radical axis and I is one of the centers hence IA is the line joining the centers).
Hint 2: We prove that A is the center of the circle . Using cosine we show that 
is a 30-60-90 triangle as IH = r and IA = 2r. This implies 
is 
. Hence it is sufficient to show 
is . A simple angle chasing in triangle 
and 
completes the proof (we observe that 
and similarly with the other angles).
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