Let ”ABC” be a triangle, ”I” its in-centre; A_1, B_1, C_1 be the reflections of I in BC, CA, AB respectively. Suppose the circumcircle of triangle A_1 B_1 C_1 passes through A. Prove that B_1, C_1, I, I_1 are concyclic, where I_1 is the incentre of triangle A_1 B_1 C_1 .

Basic Sketch:

INMO 2008 Problem 1Hint 1: I is indeed the circumcenter of A A_1 B_1 C_1 with circum radius = 2r. B_1 C_1 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 B_1, C_1, I, I_1 . Using cosine we show that \Delta AIH is a 30-60-90 triangle as IH = r and IA = 2r. This implies \angle B_1 I C_1 is 120^o . Hence it is sufficient to show \angle B_1 I_1 C_1 is 120^o . A simple angle chasing in triangle A_1 I_1 C_1 and B_1 I_1 C_1 completes the proof (we observe that \angle A_1 C_1 B_1 = \frac {A+B} {2} and similarly with the other angles).

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