A straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.
与有理中面线段共度的线段本身也是有理中面线段。
本页以“有理中面线段的共度线段”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 a rational plus a medial area, and let CD be commensurable with AB; it is to be proved that CD is also the side of a rational plus a medial area.
设AB是有理中面线段，在E点分割，则AE、EB是平方不可通约的线段，且它们上的正方形之和是中面，而它们所成矩形是有理的。
Let AB 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, but the rectangle contained by them rational.
设CD与AB共度，作相同的构造，可类似证明CF、FD平方不可通约。
[X. 40] Let the same construction be made as before.
由于AE、EB上的正方形之和与CF、FD上的正方形之和共度，且AE、EB所成矩形与CF、FD所成矩形共度。
We can then prove similarly that CF, FD are incommensurable in square, and 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, and the rectangle CF, FD rational.
因此CF、FD上的正方形之和也是中面，而它们所成矩形是有理的，故CD是有理中面线段。