Euclid 1-43 Parallelogram Area Theorem

If ABCD is a parallelogram, and E is an arbitrary point on diagonal AC, then the areas of parallelograms EFBG and EHDI are equal.

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Euclid 1-43 Parallelogram Area Theorem

Can you prove the result? This result appears in Euclid's Elements (300 BC) and the original proof can be found at Book 1, Proposition 43

Note: The converse of the result is also true. If parallelograms AFEI, FBGE, EGCH and IEHD are constructed so that the areas of FBGE and IEHD are equal, then E lies on the diagonal AC. Can you also prove this result?

Heron of Alexandria proved the converse and used it (together with some others) to prove the concurrency problem given at Bride's Chair Concurrency. [Reference: Hawking, S. (Ed.). (2006). God Created the Integers: Mathematical Breakthroughs that changed History. Penguin Books, pp.23-24.]

Michael de Villiers, 17 Jan 2010.