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rhombus incribed in triangle, 3 circumenters and an intersecion collinear

Source: 2009 Sharygin Geometry Olympiad Final Round problem 5 grade 10

July 26, 2018
geometryrhombuscollinearCircumcenter

Problem Statement

Rhombus CKLNCKLN is inscribed into triangle ABCABC in such way that point LL lies on side ABAB, point NN lies on side ACAC, point KK lies on side BCBC. O1,O2O_1, O_2 and OO are the circumcenters of triangles ACL,BCLACL, BCL and ABCABC respectively. Let PP be the common point of circles ANLANL and BKLBKL, distinct from LL. Prove that points O1,O2,OO_1, O_2, O and PP are concyclic.
(D.Prokopenko)
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