Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

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the sixth binomial, the “side” of the area is the side of the sum of two medial areas; [X. 59] so that the “side” of the area AD is also the side of the sum of two medial areas.

Therefore etc. Q. E. D.

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PROPOSITION 73.

If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square only, the remainder is irrational; and let it be called an apotome.

For from the rational straight line AB let the rational straight line BC, commensurable with the whole in square only, be subtracted; I say that the remainder AC is the irrational straight line called apotome.

For, since AB is incommensurable in length with BC, and, as AB is to BC, so is the square on AB to the rectangle AB, BC, therefore the square on AB is incommensurable with the rectangle AB, BC. [X. 11]

But the squares on AB, BC are commensurable with the square on AB, [X. 15] and twice the rectangle AB, BC is commensurable with the rectangle AB, BC. [X. 6]