(41C) Area of Triangle (SSS)

[Albert Chan]

(11-16-2018 03:33 AM)Albert Chan Wrote:  Trivia: Area Δ = √((ab-y)*y), max(y/(ab)) = 1/4

Simpler proof, using Law of Cosine: c² = (a-b)² + 4ab sin(C/2)²

Angle A ≥ B ≥ C → max(C) = 60° → y/(ab) = sin(C/2)² ≤ ¼

(11-12-2018 08:28 PM)Dieter Wrote:  Both the formula and the program look good.
But is there a proof that the formula is (at least) as exact as Kahan's?

This is Kahan's formula: Area Δ = ¼ √((a+(b+c)) (c-(a-b)) (c+(a-b)) (a+(b-c))

Note that 4y = (c-(a-b)) (c+(a-b)) = middle 2 terms inside Kahan's √

All is needed is to show 4(ab-y) also as accurate as (a+(b+c)) (a+(b-c))

With 0 < y/(ab) ≤ ¼, (ab-y) terms are too far apart to hit by catastrophic cancellation.