A point is chosen along the diagonal of a parallelogram, and segments parallel to the sides are constructed through that point. This applet allows you to investigate an interesting phenomenon regarding the areas of the four smaller parallelograms formed by this construction.
Point B can be moved to change the configuration of the parallelogram. (However, points A, C, and D are fixed points.)
Use the Show Diagonal buttons to highlight one of the diagonals. An arbitrary point P is chosen on the diagonal, and two segments parallel to the sides are constructed through P. The areas of the four resulting parallelograms are displayed below the figure.
The position of point P can be changed by dragging it along the diagonal.
Adjust the size and shape of the parallelogram by dragging point B. Then, use the Show Diagonal button to make one of the diagonals appear, and move point P to any location.
Look at the areas shown below the figure. What do you notice?
Try the same exploration with the other diagonal. Do the same relationships appear to hold?
Formulate a conjecture based on your observations. Can you prove (or disprove) your conjecture?
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