Euclid's Elements (Jihe Yuanben), Vol 6

They also lie between the same parallel lines. Therefore, the ratio of the segments AD original: 甲丁 to DB original: 丁乙, and the ratio of AE original: 甲戊 to EC original: 戊丙, are both equal to the ratio of triangle AED original: 甲戊丁 to triangle DEB original: 丁戊乙, or to triangle DEC original: 丁戊丙 since triangle DEB and triangle DEC are equal, then AD to DB is also equal to AE to EC Volume 51.

Second explanation: Inside triangle ABC original: 甲乙丙, a line DE original: 丁戊 divides AB original: 甲乙 and AC original: 甲丙 at points D original: 丁 and E original: 戊 into equal proportions. The proposition states that DE original: 丁戊 and BC original: 乙丙 are parallel lines.

Proof: Construct line segments DC original: 丁丙 and EB original: 戊乙. The ratio of the bases AD original: 甲丁 to DB original: 丁乙 is equal to the ratio of triangle AED original: 甲戊丁 to triangle DEB original: 丁戊乙 because they lie between parallel lines, see this volume, Proposition 1. And...

Elements of Geometry