RU-ALLRUSSIAN 1968 Problem 8

Given point O inside the acute- angled triangle ABC, and point O' inside the acute- angled triangle A'B'C'. D, E, F are the feet of the perpendiculars from O to BC, CA, AB respectively, and D', E', F' are the feet of the perpendiculars from O' to B'C', C'A', A'B' respectively. OD is parallel to O'A', OE is parallel to O'B' and OF is parallel to O'C'. Also OD- O'A' = OE- O'B' = OF- O'C'. Prove that O'D' is parallel to OA, O'E' to OB and O'F' to OC, and that O'D' OA = O'E' OB = O'F' OC.