## 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.

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.]

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Michael de Villiers, 17 Jan 2010.