Euclid's Elements (Jihe Yuanben), Vol 6

A geometric diagram shows two triangles, Triangle ABC original: 甲乙丙 and Triangle DEF original: 丁戊己, alongside two parallelograms, ACDB original: 甲庚乙丙 and DFEG original: 丁戊己辛. The diagram uses labels based on the Chinese Celestial Stems to denote points and lines, illustrating how shared bases and parallel heights relate to the proportions of the figures.

Explanation: Consider triangles ABC original: 甲乙丙 and DEF original: 丁戊己, and parallelograms ACDB original: 甲庚乙丙 and DFEG original: 丁戊己辛. Given that their bases BC original: 乙丙 and EF original: 戊己 are equal, the proposition states that the ratio of triangles ABC original: 甲乙丙 and DEF original: 丁戊己, as well as the ratio of parallelograms ACDB original: 甲庚乙丙 and DFEG original: 丁戊己辛, are all equal to the ratio of their heights AH original: 甲壬 and DK original: 丁癸.

Proof: Construct a line segment HI original: 子壬 equal in length to BC original: 乙丙, and another line segment KL original: 丑癸 equal in length to EF original: 戊己. Then, draw lines AI original: 甲子 and DL original: 丁丑. Triangle AHI original: 甲壬子 has the same base and height as triangle ABC original: 甲乙丙, so they are equal referencing Book 1, Proposition 38. It is also evident that DKL original: 丁癸丑 and...