elem.3.19
若一直线与圆相切，且从切点作一直线与切线垂直，则圆心位于该垂线上。
本页以“切点垂线过圆心”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。
For let a straight line DE touch the circle ABC at the point C, and from C let CA be drawn at right angles to DE; I say that the centre of the circle is on AC. For suppose it is not, but, if possible, let F be the centre, and let CF be joined.
设直线DE与圆ABC切于点C，从C作CA垂直于DE。
Since a straight line DE touches the circle ABC, and FC has been joined from the centre to the point of contact, FC is perpendicular to DE; [III. 18] therefore the angle FCE is right.
假设圆心不在AC上，而设F为圆心，连接CF。
But the angle ACE is also right; therefore the angle FCE is equal to the angle ACE, the less to the greater: which is impossible. Therefore F is not the centre of the circle ABC.
因DE切圆ABC，且FC为从圆心到切点的连线，故FC垂直于DE，角FCE为直角。
Similarly we can prove that neither is any other point except a point on AC.
但角ACE也是直角，因此角FCE等于角ACE，小角等于大角，矛盾。故圆心必在AC上。