« Vrati se
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.

Slični zadaci

#NaslovOznakeRj.KvalitetaTežina
1976IMO Shortlist 1997 problem 201
1974IMO Shortlist 1997 problem 183
1965IMO Shortlist 1997 problem 94
1219IMO Shortlist 1966 problem 360
1216IMO Shortlist 1966 problem 330
1211IMO Shortlist 1966 problem 281