elem.3.5
如果两个圆彼此相交，则它们不会有相同的圆心。
本页以“相交两圆不同心”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。
For let the circles ABC, CDG cut one another at the points B, C; I say that they will not have the same centre. For, if possible, let it be E; let EC be joined, and let EFG be drawn through at random.
设圆ABC和圆CDG相交于点B和C。假设它们有相同的圆心E。
Then, since the point E is the centre of the circle ABC, EC is equal to EF.
连接EC，并任意作直线EFG。
[I. Def. 15] Again, since the point E is the centre of the circle CDG, EC is equal to EG. But EC was proved equal to EF also; therefore EF is also equal to EG, the less to the greater : which is impossible.
因为E是圆ABC的圆心，所以EC等于EF。又因为E是圆CDG的圆心，所以EC等于EG。
Therefore the point E is not the centre of the circles ABC, CDG.
因此EF等于EG，即较小者等于较大者，这是不可能的。所以E不是两圆的共同圆心。