MathNet 2013 Question 40

Triangle $ABC$ has circumcenter $O$ and incenter $I$. Points $D,E,F$ lie respectively on $BC,CA,AB$ and satisfy $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of triangles $BFD$ and $CDE$ intersect at $D,P$. Prove that $OP=OI$.