If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles will be in a straight line.
若两个三角形有两边分别成比例，且将它们拼合于一个角，使得对应边平行，则这两个三角形的第三边在同一直线上。
两三角形 ABC、DCE 共顶点 C：若 ∠ACB 与 ∠DCE 相加为一直角（即 BCE 共线），且 BA:AC=CD:DE，则 AC 与 CE 也共线。
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Let ABC, DCE be two triangles having the two sides BA, AC proportional to the two sides DC, DE, so that, as AB is to AC, so is DC to DE, and AB parallel to DC, and AC to DE; I say that BC is in a straight line with CE. For, since AB is parallel to DC, and the straight line AC has fallen upon them, the alternate angles BAC, ACD are equal to one another.
设三角形ABC和DCE中，BA比AC等于DC比DE，且AB平行于DC，AC平行于DE。
[I. 29] For the same reason the angle CDE is also equal to the angle ACD; so that the angle BAC is equal to the angle CDE. And, since ABC, DCE are two triangles having one angle, the angle at A, equal to one angle, the angle at D, and the sides about the equal angles proportional, so that, as BA is to AC, so is CD to DE, therefore the triangle ABC is equiangular with the triangle DCE; [VI. 6] therefore the angle ABC is equal to the angle DCE.
由平行线性质，内错角BAC等于ACD，同理角CDE等于角ACD，故角BAC等于角CDE。
But the angle ACD was also proved equal to the angle BAC; therefore the whole angle ACE is equal to the two angles ABC, BAC. Let the angle ACB be added to each; therefore the angles ACE, ACB are equal to the angles BAC, ACB, CBA.
因两三角形有一对角相等，且夹等角的边成比例，故三角形ABC与DCE等角，得角ABC等于角DCE。
But the angles BAC, ABC, ACB are equal to two right angles; [I. 32] therefore the angles ACE, ACB are also equal to two right angles. Therefore with a straight line AC, and at the point C on it, the two straight lines BC, CE not lying on the same side make the adjacent angles ACE, ACB equal to two right angles; therefore BC is in a straight line with CE.
角ACE等于角ABC与角BAC之和，加公共角ACB得两直角，故BC与CE在同一直线上。