Dear Alex,

Thank you very much for your interesting reference! You are right.

Following Euclid I.43 we can rearrange this fact as following:

From any point inside a parallelogram we construct three lines: two lines parallel with the sides of the parallelogram and one line parallel with one diagonal.
Two lines parallel with the sides divide original parallelogram into four small parallelograms. If two small parallelograms have not any vertex on one diagonal we name they as complement parallelograms with respect to this diagonal.
One line parallel with one diagonal bound with this diagonal and two parallel sides of original parallelogram one new small parallelogram. We name this parallelogram as diagonal parallelogram. Of course there are two diagonal parallelograms with given diagonal but easy to show that they are always area equal.
Now we can formulate generalized Euclid I.43 as following:
In any parallelogram two complement parallelograms of given inside point with respect to given diagonal have area difference as area of respective diagonal parallelogram.
We get the Euclid I.43 if area of diagonal parallelogram is zero, i.e. given point is on the diagonal.
We can also formulate the fact for any point on the plane (inside and outside):
In any parallelogram three areas of two complement parallelograms and one respective diagonal parallelogram of given point with respect to given diagonal hold a property: one is sum of other two.

Thanks to Euclid we have our geometry now!
Best regards,
Bui Quang Tuan