Feynman Parallelogram Variation

For a parallelogram ABCD in the plane, if each side is divided by a point (1/p), (p >= 2) (measured say anti-clockwise), find a formula relating the area of ABCD with that of A'B'C'D' (parallelogram formed by these points).

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Feynman Parallelogram Variation: 1/2 division

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Feynman Parallelogram Variation: 1/3 division

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Feynman Parallelogram Variation: 1/4 division

Can you generalize even further for sides divided into different ratios?


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Michael de Villiers, Sept 2009.