Euclid's Elements (Jihe Yuanben), Vol 6

...triangle DEF original: 丁戊己 are also equal referencing Book 1, Proposition 38. Therefore, the ratio of ABC original: 甲乙丙 to DEF original: 丁戊己 is the same as the ratio of AHI original: 甲壬子 to DKL original: 丁癸丑 referencing Book 5, Proposition 7. Now, taking AI original: 甲壬子 and DL original: 丁癸丑 as the bases, the ratio of the two triangles AHI original: 甲壬子 and DKL original: 丁癸丑 is equal to the ratio of their bases AH original: 甲壬 and DK original: 丁癸 referencing this volume, Proposition 1. The ratio of ABC original: 甲乙丙 to DEF original: 丁戊己 is likewise equal to the ratio of AH original: 甲壬 to DK original: 丁癸. Furthermore, since the ratio of the two triangles ABC original: 甲乙丙 and DEF original: 丁戊己 is increased by the same factor, it equals the ratio of the two parallelograms ACDB original: 甲庚乙丙 and DFEG original: 丁戊己辛 referencing Book 5, Proposition 15. Thus, the ratio of the two parallelograms is also the same as the ratio of AH original: 甲壬 to DK original: 丁癸 referencing Book 15. If one constructs the lines GZ original: 庚子 and SY original: 辛丑, the same logic follows.