Question

You can calculate triangle area when you know all three sides by using Heron's Formula. You can also use a formula discovered by the Chinese which I found in Wikipedia. The formula does work (but I don't know what they mean by a>=b>=c). My question is, does this formula have a name?
https://en.wikipedia.org/wiki/Heron%27s_formula

Answer 1

a ⩾ b ⩾ c, simply means, "a" and "b" may well be either equal, but if either will be larger than the other, it has to be "a", and "b" and "c" may well be equal, but if either has to be larger it has to be "b", and by associative property, "a" is larger than "c" as well, or equal.

does it have a name?  no that I can think of offhand.  Bear in mind that you can simplify or expand or derive an expression for a certain case or scenario and it doesn't have to have a name, in fact, deriving an equation is done quite a bit in math, and it doesn't have to have a name, but one can give it a fitting one.

Answer 2

Answer:

It is a rearrangement of Heron's formula

Step-by-step explanation:

Straightforward expansion of Heron's formula will result in a set of terms that can be arranged to the form of your second formula.

Interestingly, the trig formula for the area of a triangle can also be used to produce Heron's formula and/or your Chinese formula, using the Law of Cosines to relate side lengths to angle.

Area = (1/2)ab·sin(C)

cos(C) = (a^2+b^2-c^2)/(2ab)

sin(C) = √(1 -cos(C)^2)

_____

The requirement to order a, b, c from longest to shortest should not be necessary. If the triangle inequality is met, the product under the radical in Heron's formula will always be positive. Since the other formula is simply a rearrangement of that, the same should be true there.