If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle.
在三角形内部取一点并连到两端点，所得两线小于原两边之和，但所夹角大于原角。
三角形 ABC，D 在内部，连 BD、DC；BD 延长至交 AC 于 E。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
On BC, one of the sides of the triangle ABC, from its extremities B, C, let the two straight lines BD, DC be constructed meeting within the triangle; I say that BD, DC are less than the remaining two sides of the triangle BA, AC, but contain an angle BDC greater than the angle BAC. For let BD be drawn through to E. Then, since in any triangle two sides are greater than the remaining one, [I. 20] therefore, in the triangle ABE, the two sides AB, AE are greater than BE.
在三角形 ABC 内取点 D，连接 DB、DC。
Let EC be added to each; therefore BA, AC are greater than BE, EC. Again, since, in the triangle CED, the two sides CE, ED are greater than CD, let DB be added to each; therefore CE, EB are greater than CD, DB.
分别在三角形 ABD 与 ACD 中使用 euclid-elements/book1-prop-020。
But BA, AC were proved greater than BE, EC; therefore BA, AC are much greater than BD, DC. Again, since in any triangle the exterior angle is greater than the interior and opposite angle, [I. 16] therefore, in the triangle CDE, the exterior angle BDC is greater than the angle CED. For the same reason, moreover, in the triangle ABE also, the exterior angle CEB is greater than the angle BAC.
相加后得到 DB+DC 小于 AB+AC。
But the angle BDC was proved greater than the angle CEB; therefore the angle BDC is much greater than the angle BAC. Therefore etc.
角 BDC 作为相关外角链的一部分，大于原角 BAC。