What is Helen's formula?

Given three sides a, b and c of a triangle, then:

(Helen formula) (p=(a+b+c)/2)

S=sqrt[p(p-a)(p-b)(p-c)]

= sqrt[( 1/ 16)(a+b+c)(a+b-c)(a+c-b)(b+ c-a)]

= 1/4 sqrt[(a+b+c)(a+b-c)(a+c-b)(b+c-a)]

Extended data:

Mathematics in ancient Greece developed to Alexandria, and its application was greatly developed. The most prominent point is the development of trigonometry. In the process of solving a triangle, a difficult problem is how to directly calculate the area of the triangle by using its three sides.

This formula was drawn by Archimedes, an ancient Greek mathematician, but people often named it after Helen, an ancient Greek mathematician, because this formula first appeared in Geodesy in the Sea and was proved in Helen's book Measuring Instruments and Metrics.

Qin Zai 1247, a mathematician in the Song Dynasty in China, independently put forward the "triclinic quadrature". Although it is different from Helen's formula in form, it is completely equivalent to Helen's formula, which fills a blank in the history of Chinese mathematics, from which we can see that China's ancient mathematics level is very high.