elem.3.18
若一直线与圆相切，且从圆心到切点连一直线，则该连线垂直于切线。
本页以“切线与半径垂直定理”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, let the centre F of the circle ABC be taken, and let FC be joined from F to C; I say that FC is perpendicular to DE. For, if not, let FG be drawn from F perpendicular to DE.
设直线DE切圆ABC于点C，取圆心F，连接FC。
Then, since the angle FGC is right, the angle FCG is acute; [I. 17] and the greater angle is subtended by the greater side; [I. 19] therefore FC is greater than FG.
假设FC不垂直于DE，则从F作FG垂直于DE。
But FC is equal to FB; therefore FB is also greater than FG, the less than the greater: which is impossible. Therefore FG is not perpendicular to DE.
由于角FGC为直角，角FCG为锐角，故大边对大角，得FC大于FG。
Similarly we can prove that neither is any other straight line except FC; therefore FC is perpendicular to DE.
但FC等于FB，故FB也大于FG，矛盾。因此FC垂直于DE。