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 .
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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