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R,P,D and S are concyclic

Source: Spanish MO 2012 Q6

June 7, 2012
circumcirclegeometry proposedgeometry

Problem Statement

Let ABCABC be an acute-angled triangle. Let ω\omega be the inscribed circle with centre II, Ω\Omega be the circumscribed circle with centre OO and MM be the midpoint of the altitude AHAH where HH lies on BCBC. The circle ω\omega be tangent to the side BCBC at the point DD. The line MDMD cuts ω\omega at a second point PP and the perpendicular from II to MDMD cuts BCBC at NN. The lines NRNR and NSNS are tangent to the circle Ω\Omega at RR and SS respectively. Prove that the points R,P,DR,P,D and SS lie on the same circle.
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