MathDB
MMO 175 Moscow MO 1950 n circles concurrent given, points coincide wanted

Source:

August 6, 2019
geometrycirclesconcurrentcoincides

Problem Statement

a) We are given nn circles O1,O2,...,OnO_1, O_2, . . . , O_n, passing through one point OO. Let A1,...,AnA_1, . . . , A_n denote the second intersection points of O1O_1 with O2,O2O_2, O_2 with O3O_3, etc., OnO_n with O1O_1, respectively. We choose an arbitrary point B1B_1 on O1O_1 and draw a line segment through A1A_1 and B1B_1 to the second intersection with O2O_2 at B2B_2, then draw a line segment through A2A_2 and B2B_2 to the second intersection with O3O_3 at B3B_3, etc., until we get a point BnB_n on OnO_n. We draw the line segment through BnB_n and AnA_n to the second intersection with O1O_1 at Bn+1B_{n+1}. If BkB_k and AkA_k coincide for some kk, we draw the tangent to OkO_k through AkA_k until this tangent intersects Ok+1O_{k+1} at Bk+1B_{k+1}. Prove that Bn+1B_{n+1} coincides with B1B_1.
b) for n=3n=3 the same problem.
Back to ProblemsView on AoPS