In any triangle two sides taken together in any manner are greater than the remaining one.
任一三角形中，任意两边合起来大于第三边。
三角形 ABC；把 BA 延长至 D 使 AD=AC；连 DC，由等腰三角形性质比较角。
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 ABC be a triangle; I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely BA, AC greater than BC, AB, BC greater than AC, BC, CA greater than AB. For let BA be drawn through to the point D, let DA be made equal to CA, and let DC be joined.
延长一边，并在延长线上取一段等于另一边。
Then, since DA is equal to AC, the angle ADC is also equal to the angle ACD; [I. 5] therefore the angle BCD is greater than the angle ADC. [C.N. 5] And, since DCB is a triangle having the angle BCD greater than the angle BDC, and the greater angle is subtended by the greater side, [I. 19] therefore DB is greater than BC.
由等腰三角形底角相等和 euclid-elements/book1-prop-016，得到外部辅助三角形中的角大小关系。
But DA is equal to AC; therefore BA, AC are greater than BC. Similarly we can prove that AB, BC are also greater than CA, and BC, CA than AB.
再用 euclid-elements/book1-prop-019，把角大小关系转回边大小关系。
Therefore etc.
得到两边之和大于第三边；换取任意两边同理。