Euclid's Elements (Jihe Yuanben), Vol 6

The angle Jia is common to both, so the two shapes are similar, and the ratios of the sides adjacent to the equal angles are equal.

Current Proposition

Addition: If a line is drawn parallel to any side within a triangle, and a point is taken on this side from which a line is drawn to the opposite angle, the two resulting segments will have proportional ratios.

A geometric diagram shows a triangle with vertices labeled Jia (top), Yi (bottom left), and Bing (bottom right). A horizontal line segment Ding-Wu is drawn parallel to the base Yi-Bing. A vertical line segment is drawn from vertex Jia through the midpoint Geng of Ding-Wu to the midpoint Ji of Yi-Bing.

Explanation: Within triangle Jia-Yi-Bing, draw line Ding-Wu parallel to Yi-Bing. Then, take an arbitrary point Ji on the side Yi-Bing and draw a line to angle Jia, which divides Ding-Wu at Geng. The proposition states that the ratio of Yi-Ji to...