Problem

Source: Brazil Ibero TST 2020 #3

Tags: geometry, incenter, projective geometry, geometric transformation, inscribed quadrilateral



Let ABCD be a quadrilateral with a incircle ω. Let I be the center of ω, suppose that the lines AD and BC intersect at Q and the lines AB and CD intersect at P with B is in the segment AP and D is in the segment AQ. Let X and Y the incenters of PBD and QBD respectively. Let R be the intersection of PY and QX. Prove that the line IR is perpendicular to BD.