Euclid's Elements (Jihe Yuanben), Vol 6

Proof: Following the previous method, construct line BE original: 乙戊 parallel to AD original: 甲丁 and extend line AC original: 丙甲 to E original: 戊. Since the ratio of AB original: 乙甲 to AC original: 甲丙 is the same as the ratio of BD original: 乙丁 to DC original: 丁丙, and line AD original: 甲丁 is parallel to side BE original: 戊乙, the ratio of BD original: 乙丁 to DC original: 丁丙 is the same as the ratio of AE original: 戊甲 to AC original: 甲丙 see Proposition 2 of this volume. Therefore, the ratio of AB original: 乙甲 to AC original: 甲丙 is also the same as the ratio of AE original: 戊甲 to AC original: 甲丙, Book 5, Proposition 11 which means the two lines AE original: 戊甲 and AB original: 乙甲 are equal. Book 5, Proposition 9 Thus, angle AEB original: 甲乙戊 is also equal to angle AEB original: 戊. Book 1, Proposition 5 Since angle AEB original: 甲乙戊 and angle BAD original: 乙甲丁 are interior opposite angles of parallel lines and are equal, and the exterior angle DAC original: 丁甲丙 is equal to the interior angle AEB original: 戊, Book 1, Proposition 29 then angle BAD original: 乙甲丁 and angle DAC original: 丁甲丙 must be equal.

A geometric diagram shows a triangle 乙戊丙. A line segment 甲丁 is drawn within the triangle, intersecting the side 戊丙 at point 甲 and the base 乙丙 at point 丁. The vertices and intersection points are labeled with Celestial Stems: 戊 at the top vertex, 乙 at the bottom left, 丙 at the bottom right, 甲 on the right side, and 丁 on the base.