Euclid's Elements (Jihe Yuanben), Vol 6

The proposition states that the ratio of AD original: 甲丁 to DB original: 丁乙 is equal to the ratio of AE original: 甲戊 to EC original: 戊丙.

Proof: Construct line segments DC original: 丁丙 and EB original: 戊乙. Triangle DEB original: 丁戊乙 and triangle DEC original: 丁戊丙 share the same base DE original: 丁戊 and lie between the same parallel lines, so they are equal referencing Book 1, Proposition 37. The ratio of triangle AED original: 甲戊丁 to triangle DEB original: 丁戊乙 is the same as the ratio of triangle AED original: 甲戊丁 to triangle DEC original: 丁戊丙 referencing Book 5, Proposition 7. Since both triangle AED original: 甲戊丁 and triangle DEB original: 丁戊乙 lie between the same parallel lines if one draws a line through point E parallel to AB, both shapes lie within those bounds, the ratio of triangle AED original: 甲戊丁 to triangle DEB original: 丁戊乙 is equal to the ratio of their bases AD original: 甲丁 and DB original: 丁乙 referencing this volume, Proposition 1. It is also evident that the ratio of the bases AE original: 甲戊 to EC original: 戊丙 is the same as the ratio of triangle AED original: 甲戊丁 to triangle DEC original: 丁戊丙 the two triangles.