(41C) Area of Triangle (SSS)

[Albert Chan]

I derived triangle area (SSS) formula from Laws of Cosine.
Assuming c is the shortest side, this is the formula:

Let y = (c - (a-b))*(c + (a-b)) / 4, then Area Δ = √((ab-y)*y)

Prove:

c² = a² + b² - 2ab cos(C)
= (a-b)² + 2ab*(1 - cos(C))
= (a-b)² + 4ab*sin(C/2)²

let x = sin(C/2)², so 4abx = c² - (a-b)²
let y = abx, so y = (c - (a-b))*(c + (a-b)) / 4

sin(C) = √(1 - cos(C)²) = √(1 - (1-2x)²) = √(4x - 4x²) = 2√(x - x²)
Area Δ = ½ ab sin(C) = ab √(x - x²) = √((ab)(abx) - (abx)²) = √((ab-y)*y)

Above also proved Heron's formula, since y=(s-a)(s-b):
ab-y = ab - (s² - (a+b)s + ab) = -s² + (2s-c)s = s(s-c)

Area Δ = √((ab-y)*y) = √(s(s-a)(s-b)(s-c))

Update Oct 4,2022: perhaps a simpler proof

Area Δ = ½ ab sin(C) = y cot(C/2) = y √(csc(C/2)^2-1) = y √(ab/y-1) = √((ab-y)*y)