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PROPOSITION 93.

If an area be contained by a rational straight line and a third apotome, the “side” of the area is a second apotome of a medial straight line.

For let the area AB be contained by the rational straight line AC and the third apotome AD; I say that the “side” of the area AB is a second apotome of a medial straight line.

For let DG be the annex to AD; therefore AG, GD are rational straight lines commensurable in square only, and neither of the straight lines AG, GD is commensurable in length with the rational straight line AC set out, while the square on the whole AG is greater than the square on the annex DG by the square on a straight line commensurable with AG. [X. Deff. III. 3]

Since then the square on AG is greater than the square on GD by the square on a straight line commensurable with AG, therefore, if there be applied to AG a parallelogram equal to the fourth part of the square on DG and deficient by a square figure, it will divide it into commensurable parts. [X. 17]

Let then DG be bisected at E, let there be applied to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF, FG.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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