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IMO Shortlist 2014 G4

Source:

July 11, 2015
IMO Shortlistgeometry

Problem Statement

Consider a fixed circle Γ\Gamma with three fixed points A,B,A, B, and CC on it. Also, let us fix a real number λ∈(0,1)\lambda \in(0,1). For a variable point P∉{A,B,C}P \not\in\{A, B, C\} on Γ\Gamma, let MM be the point on the segment CPCP such that CM=λ⋅CPCM =\lambda\cdot CP . Let QQ be the second point of intersection of the circumcircles of the triangles AMPAMP and BMCBMC. Prove that as PP varies, the point QQ lies on a fixed circle.
Proposed by Jack Edward Smith, UK
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