elem.3.11
若两圆内切，则连接两圆圆心的直线延长后必过切点。
本页以“内切圆圆心连线过切点”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 internally at the point A, and let the centre F of the circle ABC, and the centre G of ADE, be taken; I say that the straight line joined from G to F and produced will fall on A. For suppose it does not, but, if possible, let it fall as FGH, and let AF, AG be joined.
设圆ABC与圆ADE内切于点A，圆心分别为F和G。
Then, since AG, GF are greater than FA, that is, than FH, let FG be subtracted from each; therefore the remainder AG is greater than the remainder GH.
假设连线FG延长后不过A，而交圆于H，连接AF、AG。
But AG is equal to GD; therefore GD is also greater than GH, the less than the greater: which is impossible.
由三角形两边和大于第三边，AG+GF>AF，即AG+GF>FH，两边减去FG得AG>GH。
Therefore the straight line joined from F to G will not fall outside; therefore it will fall at A on the point of contact.
但AG等于GD，故GD>GH，矛盾。因此连线必过切点A。