Prove Qin Jiushao's "triclinic product" formula and Helen's formula, and find the area of ​​the three sides of a given triangle. Three quotes from your last tweet. How to find the area of ​​a triangle directly from its three sides a, b, and c? Readers are invited to try to use th

Verify Qin Jiushao "triclinic product" formula and Helen's formula , find the area of ​​the three sides of a given triangle.

quotes three sentences from the previous tweet.

How to find the area of ​​a triangle directly from the three sides a, b, and c of the triangle?

Please try to use the cosine theorem to derive Helen's formula and Qin Jiushao's "triclinic product" formula.

Although proving Heron's formula is an after-school exercise in the textbook, many students have not tried to prove Heron's formula in three years of high school.

two, textbook exercises, broaden exploration, please try to complete:

three,

my country's Southern Song Dynasty The "triclinic product" formula discovered by the famous mathematician Qin Jiushao (about 1202-1261) is equivalent to Helen's formula.
There is a question in Volume 5 "Field Category" of Qin Jiushao's "Nine Chapters of Mathematics" : "There is a section of Shatian with three slopes. The small slope is 13 miles, the middle slope is 14 miles, and the big slope is 13 miles. Diagonal: 15 miles. Three hundred steps. "To know the geometry of the field," this question actually involves finding the area of ​​the triangle if the lengths of its three sides are known. The method for calculating in "Nine Chapters of Mathematics" is: "take the small slope power and the large slope power and subtract the middle slope power, and multiply the remainder by half. Multiply the small slope power by the large slope power and subtract it. The remainder is about four It is real. The first is to calculate the product by taking the square root. "" refers to the "triclinic product" formula.

proves Qin Jiushao's "triclinic product" formula and Helen's formula: