elem.6.8
在直角三角形中，从直角顶点向底边作垂线，则垂线两侧的三角形分别与原三角形相似，且它们彼此相似。
直角三角形 ABC，A 为直角顶点；从 A 向斜边 BC 作垂线，垂足 D 把 △ABC 分成两个与原三角形相似的子三角形 ABD、ACD。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let ABC be a right-angled triangle having the angle BAC right, and let AD be drawn from A perpendicular to BC; I say that each of the triangles ABD, ADC is similar to the whole ABC and, further, they are similar to one another. For, since the angle BAC is equal to the angle ADB, for each is right, and the angle at B is common to the two triangles ABC and ABD, therefore the remaining angle ACB is equal to the remaining angle BAD; [I. 32] therefore the triangle ABC is equiangular with the triangle ABD. Therefore, as BC which subtends the right angle in the triangle ABC is to BA which subtends the right angle in the triangle ABD, so is AB itself which subtends the angle at C in the triangle ABC to BD which subtends the equal angle BAD in the triangle ABD, and so also is AC to AD which subtends the angle at B common to the two triangles.
设直角三角形ABC，直角为BAC，从A向BC作垂线AD。
[VI. 4] Therefore the triangle ABC is both equiangular to the triangle ABD and has the sides about the equal angles proportional. Therefore the triangle ABC is similar to the triangle ABD. [VI. Def. 1] Similarly we can prove that the triangle ABC is also similar to the triangle ADC; therefore each of the triangles ABD, ADC is similar to the whole ABC.
由于角BAC等于角ADB（均为直角），且角B公共，故三角形ABC与ABD等角，从而对应边成比例，因此三角形ABC相似于ABD。
I say next that the triangles ABD, ADC are also similar to one another. For, since the right angle BDA is equal to the right angle ADC, and moreover the angle BAD was also proved equal to the angle at C, therefore the remaining angle at B is also equal to the remaining angle DAC; [I. 32] therefore the triangle ABD is equiangular with the triangle ADC. Therefore, as BD which subtends the angle BAD in the triangle ABD is to DA which subtends the angle at C in the triangle ADC equal to the angle BAD, so is AD itself which subtends the angle at B in the triangle ABD to DC which subtends the angle DAC in the triangle ADC equal to the angle at B, and so also is BA to AC, these sides subtending the right angles; [VI. 4] therefore the triangle ABD is similar to the triangle ADC.
同理可证三角形ABC相似于ADC。
[VI. Def. 1] Therefore etc. PORISM.
又因角BDA等于角ADC（直角），且角BAD等于角C，故三角形ABD与ADC等角，对应边成比例，因此它们相似。