Therefore D is medial. [X. 21] And since B, C are commensurable in square only, and, as B is to C, so is D to E, therefore D, E are also commensurable in square only. [X. 11] But D is medial; therefore E is also medial. [X. 23, addition] Therefore D, E are medial straight lines commensurable in square only. I say next that they also contain a medial rectangle. For since, as B is to C, so is D to E, therefore, alternately, as B is to D, so is C to E. [V. 16] But, as B is to D, so is D to A; therefore also, as D is to A, so is C to E; therefore the rectangle A, C is equal to the rectangle D, E. [VI. 16] But the rectangle A, C is medial; [X. 21] therefore the rectangle D, E is also medial. Therefore medial straight lines commensurable in square only have been found which contain a medial rectangle. Q. E. D.
Therefore D is medial. [X. 21] And since B, C are commensurable in square only, and, as B is to C, so is D to E, therefore D, E are also commensurable in square only. [X. 11] But D is medial; therefore E is also medial. [X. 23, addition] Therefore D, E are medial straight lines commensurable in square only. I say next that they also contain a medial rectangle. For since, as B is to C, so is D to E, therefore, alternately, as B is to D, so is C to E. [V. 16] But, as B is to D, so is D to A; therefore also, as D is to A, so is C to E; therefore the rectangle A, C is equal to the rectangle D, E. [VI. 16] But the rectangle A, C is medial; [X. 21] therefore the rectangle D, E is also medial. Therefore medial straight lines commensurable in square only have been found which contain a medial rectangle. Q. E. D.
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