Euclid's Elements (Jihe Yuanben), Vol 6

If a straight line bisects an angle of a triangle and also divides the opposite side into two segments, then the ratio of the two segments is the same as the ratio of the remaining two sides. Conversely, if the ratio of the two segments of the opposite side is the same as the ratio of the remaining two sides, then the angle is bisected.

A geometric diagram shows a triangle labeled with celestial stems (甲, 乙, 丙, 丁, 戊). A line extends from the top vertex (甲) to the base (乙丙) at point 丁. An auxiliary line extends from vertex 丙 to a point 戊, forming a larger triangle structure.

First explanation: In triangle ABC original: 甲乙丙, the line AD original: 甲丁 bisects the angle BAC original: 乙甲丙. The proposition states that 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.

Proof: Attempt to construct line segment BE original: 乙戊 parallel to AD original: 甲丁, then extend the line AC original: 丙甲.