elem.6.15
在两个三角形中，若它们有一个角相等，且面积相等，则相等角的两边成反比；反之，若有一个角相等且该角的两边成反比，则这两个三角形面积相等。
两等积三角形 ABC、ADE 共角于 A：CA 与 AD 共线，BA 与 AE 共线；等角两边成反比例 CA:AD = EA:AB。
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, ADE be equal triangles having one angle equal to one angle, namely the angle BAC to the angle DAE; I say that in the triangles ABC, ADE the sides about the equal angles are reciprocally proportional, that is to say, that, as CA is to AD, so is EA to AB. For let them be placed so that CA is in a straight line with AD; therefore EA is also in a straight line with AB. [I. 14] Let BD be joined.
设三角形ABC与ADE面积相等，且角BAC等于角DAE。将三角形放置使CA与AD共线，则EA与AB也共线。连接BD。
Since then the triangle ABC is equal to the triangle ADE, and BAD is another area, therefore, as the triangle CAB is to the triangle BAD, so is the triangle EAD to the triangle BAD. [V. 7] But, as CAB is to BAD, so is CA to AD, [VI. 1] and, as EAD is to BAD, so is EA to AB. [id.] Therefore also, as CA is to AD, so is EA to AB.
由于三角形ABC与ADE面积相等，且三角形BAD为公共区域，故三角形CAB与BAD的面积比等于三角形EAD与BAD的面积比。
[V. 11] Therefore in the triangles ABC, ADE the sides about the equal angles are reciprocally proportional. Next, let the sides of the triangles ABC, ADE be reciprocally proportional, that is to say, let EA be to AB as CA to AD; I say that the triangle ABC is equal to the triangle ADE. For, if BD be again joined, since, as CA is to AD, so is EA to AB, while, as CA is to AD, so is the triangle ABC to the triangle BAD, and, as EA is to AB, so is the triangle EAD to the triangle BAD, [VI. 1] therefore, as the triangle ABC is to the triangle BAD, so is the triangle EAD to the triangle BAD.
根据第六卷命题1，三角形CAB与BAD的面积比等于CA与AD的比，三角形EAD与BAD的面积比等于EA与AB的比。因此CA比AD等于EA比AB。
[V. 11] Therefore each of the triangles ABC, EAD has the same ratio to BAD. Therefore the triangle ABC is equal to the triangle EAD.
反之，若CA比AD等于EA比AB，则三角形ABC与BAD的面积比等于三角形EAD与BAD的面积比，故三角形ABC与EAD面积相等。