Euclid, Elements I 43©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 43: For every parallelogram, the complements of the parallelograms about the diameter are equal to one another.

(general diagram)

Diagrams for book 1 proposition 43, following the display, construction, and demonstration(diagram 1) Let there be a parallelogram, ABGD, its diameter, AG, (diagram 2) and let there be about AG parallelograms, EQ, ZH, and the so-called complements, BK, KD. I say that complement BE is equal to complement KD.
(diagram 3) For since ABGD is a parallelogram and AG its diameter, triangle ABG is equal to triangle AGD. (diagram 4) Again, since EQ is a parallelogram and AK is its diameter, triangle AEG is equal to triangle AQK. (diagram 5) For the same reasons, in fact, triangle KZG is also equal to KHG. (diagram 6) And so, since triangle AEK is equal to triangle AQK and KZG to KHG, triangle AEK with KHG is equal to triangle AQK with KZG. But a whole, triangle ABG, is equal to a whole, ADG. Therefore, a remainder, complement BK, is equal to a remainder, KD. Therefore, for every parallelogram, the complements of the parallelograms about the diameter are equal to one another, just what it was required to show.

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