elem.3.9
若在圆内取一点，从该点向圆上引出的相等线段多于两条，则该点即为圆心。
本页以“圆内点引等线定圆心”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 a circle and D a point within it, and from D let more than two equal straight lines, namely DA, DB, DC, fall on the circle ABC; I say that the point D is the centre of the circle ABC. For let AB, BC be joined and bisected at the points E, F, and let ED, FD be joined and drawn through to the points G, K, H, L.
设圆ABC内一点D，从D向圆上引出相等线段DA、DB、DC。
Then, since AE is equal to EB, and ED is common, the two sides AE, ED are equal to the two sides BE, ED; and the base DA is equal to the base DB; therefore the angle AED is equal to the angle BED. [I. 8] Therefore each of the angles AED, BED is right; [I. Def. 10] therefore GK cuts AB into two equal parts and at right angles.
连接AB、BC，分别取中点E、F，连接ED、FD并延长至G、K、H、L。
And since, if in a circle a straight line cut a straight line into two equal parts and at right angles, the centre of the circle is on the cutting straight line, [III. 1, Por.] the centre of the circle is on GK. For the same reason the centre of the circle ABC is also on HL.
由AE=EB、ED公共、DA=DB，得∠AED=∠BED，故∠AED和∠BED均为直角，即GK垂直平分AB。
And the straight lines GK, HL have no other point common but the point D; therefore the point D is the centre of the circle ABC.
同理，HL垂直平分BC。由于GK和HL仅交于D，且圆心必在垂直平分弦的直线上，故D为圆心。