Dear All My Friends,
Two following triangle area facts in parallelogram and quadrilateral are very simple and equivalent:

1. Given any point X inside parallelogram ABCD.
One line from X parallel with AD//BC intersect AB at M, CD at P.
One line from X parallel with AB//CD intersect BC at N, DA at Q.
Result: Area(BDX) = Abs(Area(MQX) - Area(NPX))

2. X is intersection of two diagonals in convex quadrilateral MNPQ. E and F are midpoints of MN and PQ respectively.
Result: Area(EFX) = 1/4*Abs(Area(MQX) - Area (NPX))

Proof of 1)
Suppose X is inside triangle ABD.
Easy to show Area(MNPQ) = 1/2*Area(ABCD)
Area(BDX) = Area(ABD) - Area(MBX) - Area(QDX) - 2*Area(MQX) =
1/2*Area(ABCD) - Area(MNX) - Area(PQX) - 2*Area(MQX) =
Area(MNPQ) - Area(MNX) - Area(PQX) - 2*Area(QMX) =
Area(NPX) + Area(MQX) - 2*Area(MQX) =
Area(NPX) - Area(MQX)

Proof of 2)
Suppose X is inside MEFQ.
Area(EFX) = Area(EFY) - Area(EXY) - Area(FXY) =
1/4*Area(MNPQ) - 1/4*Area(MNQ) - 1/4*Area(MPQ) =
1/4*Area(MNPQ) - 1/4*(Area(MNQ) + Area(MPQ)) =
1/4* Area(MNPQ) - 1/4*(Area(MNPQ) - Area(NPX) + Area(MQX)) =
1/4*(Area(NPX) - Area(MQX))

Please kindly give me references if any one knows about these!

Thank you and best regards,
Bui Quang Tuan