## Euclid I.43: What Is This About?

A Mathematical Droodle.

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Copyright © 1996-2017 Alexander BogomolnyIn the applet, parallelograms and triangles of the same color that lie on opposite sides of the diagonal have equal areas.

What if applet does not run? |

The applet allows up to 10 divisor points on the diagonal. The statement remains true for any number of points and follows immediately from the case of a single point. The latter, simple as it is, has been included as Proposition I.43 in Euclid's *Elements*:

In any parallelogram the complements of the parallelograms about the diameter are equal to one another. |

(This is one part of the lemma used in establishing the Theorem of Complete Quadrilateral.)

By SSS or ASA the triangles AKH, KCF and ACD are equal to the triangles KEA, CGK and CBA, respectively. Subtracting the small rectangles from the including big ones, we see that the areas of the two remaining parallelograms HKFD and EBGK are equal, although the shapes themselves are not necessarily so.

With two or more points on the diagonal, the argument must be simply repeated for various parallelograms, and equal areas are subtracted from equal areas.

(Euclid's statement admits an interesting extension.)

### References

- T. L. Heath,
*EUCLID: The Thirteen Books of The Elements, books I and II*, Dover, 1956

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