Euclid's Elements (Jihe Yuanben), Vol 6

Explanation: In the two triangles Jia-Yi-Bing and Ding-Wu-Ji, angles Yi and Wu are equal. If the ratio of Jia-Yi to Yi-Bing is as Ding-Wu to Wu-Ji, it is stated that the remaining angles, Bing and Ji, and Jia and Ding, are each equal.

Two geometric diagrams. On the right is a triangle labeled with the characters Jia (top), Yi (bottom left), and Bing (bottom right). On the left is a quadrilateral labeled with Ding (top), Wu (left), Ji (right), and Geng (bottom).

Proof: Construct angle Ji-Wu-Geng equal to angle Yi, and construct angle Geng-Ji-Wu equal to angle Bing. Let the lines Wu-Geng and Ji-Geng meet at Geng. According to the previous proof, it is evident that the figures Jia-Yi-Bing and Geng-Wu-Ji are equiangular. Thus, the ratio of Jia-Yi to Yi-Bing is as Geng-Wu to Wu-Ji Book 6, Proposition 4. Since the ratio of Jia-Yi to Yi-Bing was originally as Ding-Wu to Wu-Ji, then Geng-Wu to Wu-Ji is also as Ding-Wu to Wu-Ji Book 5, Proposition 11, and the two lines Ding-Wu and Geng-Wu must be equal Book 5, Proposition 9. Since the two sides Ding-Wu and Geng-Wu are equal, and Wu-Ji is a shared base...