To find the centre of a given circle.
给定一个圆，求作它的圆心。
本页以“求已知圆的圆心”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。
Let ABC be the given circle; thus it is required to find the centre of the circle ABC. Let a straight line AB be drawn through it at random, and let it be bisected at the point D; from D let DC be drawn at right angles to AB and let it be drawn through to E; let CE be bisected at F; I say that F is the centre of the circle ABC. For suppose it is not, but, if possible, let G be the centre, and let GA, GD, GB be joined.
在圆内任作一条线段AB，过点D平分AB。
Then, since AD is equal to DB, and DG is common, the two sides AD, DG are equal to the two sides BD, DG respectively; and the base GA is equal to the base GB, for they are radii; therefore the angle ADG is equal to the angle GDB. [I. 8] But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right; [I. Def. 10] therefore the angle GDB is right.
从D作DC垂直于AB，并延长至E，使CE被F平分。
But the angle FDB is also right; therefore the angle FDB is equal to the angle GDB, the greater to the less: which is impossible. Therefore G is not the centre of the circle ABC. Similarly we can prove that neither is any other point except F.
假设圆心不是F而是G，连接GA、GD、GB。由于AD等于DB，DG公共，且GA等于GB（半径），故三角形ADG全等于BDG，角ADG等于角GDB。
Therefore the point F is the centre of the circle ABC. PORISM.
角GDB为直角，但角FDB也是直角，因此角FDB等于角GDB，大角等于小角，矛盾。故F是圆心。