Euclid's Elements (Jihe Yuanben), Vol 6

Extend it to E original: 戊. The angle AEB original: 甲乙戊 and the angle BAD original: 乙甲丁 are the two interior opposite angles formed by parallel lines, so they are equal. The exterior angle DAC original: 丁甲丙 is equal to the interior angle AEB original: 戊. Book 1, Proposition 29 Now, if angle BAD original: 乙甲丁 is equal to angle DAC original: 丁甲丙, then angle AEB original: 甲乙戊 is also equal to angle AEB original: 戊. Thus, the two sides AE original: 甲戊 and AB original: 甲乙 are equal. Book 1, Proposition 6 Therefore, the ratio of AE original: 戊甲 to AC original: 甲丙 is the same as the ratio of AB original: 乙甲 to AC original: 甲丙. Book 5, Proposition 7 Since the ratio of AE original: 戊甲 to AC original: 甲丙 is the same as the ratio of BD original: 乙丁 to DC original: 丁丙, see Proposition 2 of this volume then the ratio of AB original: 乙甲 to AC original: 甲丙 is also the same as the ratio of BD original: 乙丁 to DC original: 丁丙. Book 5, Proposition 11

Second explanation: The ratio of BD original: 乙丁 to DC original: 丁丙 is the same as the ratio of AB original: 乙甲 to AC original: 甲丙. The proposition states that line AD original: 甲丁 bisects angle BAC original: 乙甲丙.