Euclid's Elements (Jihe Yuanben), Vol 6

Ji-Bing is as the ratio of Ding-Geng to Geng-Wu.

Proof: Since the two triangles Jia-Ji-Yi and Jia-Geng-Ding are similar from the previous corollary, the ratio of Jia-Ji to Ji-Yi is as Jia-Geng to Geng-Ding. By alternation, the ratio of Jia-Ji to Jia-Geng is as Ji-Yi to Geng-Ding Book 5, Proposition 16. It is similarly evident that the ratio of Jia-Ji to Jia-Geng is as Ji-Bing is to Geng-Wu. Thus, the ratio of Ji-Yi to Ding-Geng is also as Ji-Bing to Geng-Wu. By alternation, the ratio of Ji-Yi to Ji-Bing is as Ding-Geng to Geng-Wu Book 5, Proposition 16.

Another proof states that since triangles Jia-Ji-Yi and Jia-Geng-Ding are similar, and triangles Jia-Ji-Bing and Jia-Geng-Wu are each similar, then the ratio of Ji-Yi to Jia-Ji is as Ding-Geng to...