Based on numerical examples involving rectangles, the following conjecture is made: If P i...

Based on numerical examples involving rectangles, the following conjecture is made: If P is any point on the diagonal of parallelogram ABCD, the parallel lines through P to the sides form two parallelograms having equal areas. Try proving it (there is a simple proof that does not involve ratios or similar triangles, only areas of parts of the given parallelogram). Do you agree with the conjecture in the case of a square?