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An infinite sequence of circles

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September 22, 2010
geometrytrigonometryTrigonometric EquationscirclesPeriodic sequenceIMO Shortlist

Problem Statement

Consider a sequence of circles K1,K2,K3,K4,…K_1,K_2,K_3,K_4, \ldots of radii r1,r2,r3,r4,…r_1, r_2, r_3, r_4, \ldots , respectively, situated inside a triangle ABCABC. The circle K1K_1 is tangent to ABAB and ACAC; K2K_2 is tangent to K1K_1, BABA, and BCBC; K3K_3 is tangent to K2K_2, CACA, and CBCB; K4K_4 is tangent to K3K_3, ABAB, and ACAC; etc. (a) Prove the relation r1cot⁡12A+2r1r2+r2cot⁡12B=r(cot⁡12A+cot⁡12B)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 rr is the radius of the incircle of the triangle ABCABC. Deduce the existence of a t1t_1 such that r1=rcot⁡12Bcot⁡12Csin⁡2t1r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1 (b) Prove that the sequence of circles K1,K2,…K_1,K_2, \ldots is periodic.
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