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20

Part of 2014 Sharygin Geometry Olympiad

Problems(1)

if 4 circumcircles ae concurrent, then other 4 circumcircles ae concurrent

Source: Sharygin Geometry Olympiad 2014 Correspondence Round P20

8/1/2018
A quadrilateral KLMNKLMN is given. A circle with center OO meets its side KLKL at points AA and A1A_1, side LMLM at points BB and B1B_1, etc. Prove that if the circumcircles of triangles KDA,LAB,MBCKDA, LAB, MBC and NCDNCD concur at point PP, then a) the circumcircles of triangles KD1A1,LA1B1,MB1C1KD_1A_1, LA_1B_1, MB_1C_1 and NC1D1NC1D1 also concur at some point QQ; b) point OO lies on the perpendicular bisector to PQPQ.
geometrycircumcircleconcurrencyconcurrentcircles