A straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.
与两中项和之边可公度的线段也是两中项和之边。
本页以“两中项和之边可公度性”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let AB be the side of the sum of two medial areas, and CD commensurable with AB; it is to be proved that CD is also the side of the sum of two medial areas.
设AB为两中项和之边，CD与AB可公度。
For, since AB is the side of the sum of two medial areas, let it be divided into its straight lines at E; therefore AE, EB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and furthermore the sum of the squares on AE, EB incommensurable with the rectangle AE, EB.
将AB分为AE、EB，则AE、EB平方不可公度，且平方和与所成矩形均为中项，且平方和与矩形不可公度。
[X. 41] Let the same construction be made as before.
作相同构造，可证CF、FD平方不可公度。
We can then prove similarly that CF, FD are also incommensurable in square, the sum of the squares on AE, EB is commensurable with the sum of the squares on CF, FD, and the rectangle AE, EB with the rectangle CF, FD; so that the sum of the squares on CF, FD is also medial, the rectangle CF, FD is medial, and moreover the sum of the squares on CF, FD is incommensurable with the rectangle CF, FD.
由AE、EB平方和与CF、FD平方和可公度，矩形AE、EB与矩形CF、FD可公度，得CF、FD平方和为且矩形均为中项，且平方和与矩形不可公度，故CD为两中项和之边。