IMO Shortlist 1982 problem 14

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Dodao/la: arhiva
April 2, 2012
Let ABCD be a convex plane quadrilateral and let A_1 denote the circumcenter of \triangle BCD. Define B_1, C_1,D_1 in a corresponding way.

(a) Prove that either all of A_1,B_1, C_1,D_1 coincide in one point, or they are all distinct. Assuming the latter case, show that A_1, C1 are on opposite sides of the line B_1D_1, and similarly,B_1,D_1 are on opposite sides of the line A_1C_1. (This establishes the convexity of the quadrilateral A_1B_1C_1D_1.)

(b) Denote by A_2 the circumcenter of B_1C_1D_1, and define B_2, C_2,D_2 in an analogous way. Show that the quadrilateral A_2B_2C_2D_2 is similar to the quadrilateral ABCD.
Source: Međunarodna matematička olimpijada, shortlist 1982