elem.3.12
如果两个圆彼此外切，那么连接它们圆心的直线必经过切点。
本页以“两圆外切时圆心连线过切点”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 two circles ABC, ADE touch one another externally at the point A, and let the centre F of ABC, and the centre G of ADE, be taken; I say that the straight line joined from F to G will pass through the point of contact at A. For suppose it does not, but, if possible, let it pass as FCDG, and let AF, AG be joined.
设圆ABC和圆ADE外切于点A，圆心分别为F和G。
Then, since the point F is the centre of the circle ABC, FA is equal to FC.
假设连接F和G的直线不经过切点A，而经过点C和D，连接AF和AG。
Again, since the point G is the centre of the circle ADE, GA is equal to GD. But FA was also proved equal to FC; therefore FA, AG are equal to FC, GD, so that the whole FG is greater than FA, AG; but it is also less [I. 20]: which is impossible.
因为F是圆ABC的圆心，所以FA等于FC；同理，G是圆ADE的圆心，所以GA等于GD。
Therefore the straight line joined from F to G will not fail to pass through the point of contact at A; therefore it will pass through it.
于是FA与AG之和等于FC与GD之和，因此FG大于FA与AG之和，但根据三角形两边之和大于第三边，FG小于FA与AG之和，矛盾。故假设不成立，直线FG必经过点A。