ÖMO 2009 Problem 3

Let $P$ be the point in the interior of $\vartriangle ABC$ . Let $D$ be the intersection of the lines $AP$ and $BC$ and let $A'$ be the point such that $\overrightarrow{AD} = \overrightarrow{DA'}$ . The points $B'$ and $C'$ are defined in the similar way. Determine all points $P$ for which the triangles $A'BC, AB'C$ , and $ABC'$ are congruent to $\vartriangle ABC$ .