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IMO Shortlist 2012, Geometry 8

Source: IMO Shortlist 2012, Geometry 8

July 29, 2013
geometrycircumcircleIMO Shortlist

Problem Statement

Let ABCABC be a triangle with circumcircle ω\omega and ℓ\ell a line without common points with ω\omega. Denote by PP the foot of the perpendicular from the center of ω\omega to ℓ\ell. The side-lines BC,CA,ABBC,CA,AB intersect ℓ\ell at the points X,Y,ZX,Y,Z different from PP. Prove that the circumcircles of the triangles AXPAXP, BYPBYP and CZPCZP have a common point different from PP or are mutually tangent at PP.
Proposed by Cosmin Pohoata, Romania
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