OME 2004 Problem 2

${ABCD}$ is a quadrilateral, ${P}$ and ${Q}$ are midpoints of the diagonals ${BD}$ and ${AC}$ , respectively. The lines parallel to the diagonals originating from ${P}$ and ${Q}$ intersect in the point ${O}$ . If we join the four midpoints of the sides, ${X}$ , ${Y}$ , ${Z}$ , and ${T}$ , to ${O}$ , we form four quadrilaterals: ${OXBY}$ , ${OYCZ}$ , ${OZDT}$ , and ${OTAX}$ . Prove that the four newly formed quadrilaterals have the same areas.