IMO Shortlist 1972 problem 11


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Consider a sequence of circles K_1,K_2,K_3,K_4, \ldots of radii r_1, r_2, r_3, r_4, \ldots , respectively, situated inside a triangle ABC. The circle K_1 is tangent to AB and AC; K_2 is tangent to K_1, BA, and BC; K_3 is tangent to K_2, CA, and CB; K_4 is tangent to K_3, AB, and AC; etc.
(a) Prove the relation
r_1  \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2  \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right)
where r is the radius of the incircle of the triangle ABC. Deduce the existence of a t_1 such that
r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1
(b) Prove that the sequence of circles K_1,K_2, \ldots is periodic.
Izvor: Međunarodna matematička olimpijada, shortlist 1972