(41C) Area of Triangle (SSS)
12-04-2019, 03:03 PM
Post: #13
 Albert Chan Senior Member Posts: 2,697 Joined: Jul 2018
RE: (41C) Area of Triangle (SSS)
(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.
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 Messages In This Thread (41C) Area of Triangle (SSS) - Gamo - 11-10-2018, 12:20 PM RE: (41C) Area of Triangle (SSS) - Albert Chan - 11-10-2018, 03:22 PM RE: (41C) Area of Triangle (SSS) - Dieter - 11-10-2018, 08:53 PM RE: (41C) Area of Triangle (SSS) - Albert Chan - 11-12-2018, 02:55 PM RE: (41C) Area of Triangle (SSS) - Dieter - 11-12-2018, 08:28 PM RE: (41C) Area of Triangle (SSS) - Albert Chan - 11-12-2018, 10:33 PM RE: (41C) Area of Triangle (SSS) - Gamo - 11-11-2018, 05:04 AM RE: (41C) Area of Triangle (SSS) - Dieter - 11-11-2018, 07:53 AM RE: (41C) Area of Triangle (SSS) - Gamo - 11-11-2018, 12:23 PM RE: (41C) Area of Triangle (SSS) - Dieter - 11-11-2018, 04:25 PM RE: (41C) Area of Triangle (SSS) - Albert Chan - 11-12-2018, 01:32 AM RE: (41C) Area of Triangle (SSS) - Albert Chan - 11-16-2018, 03:33 AM RE: (41C) Area of Triangle (SSS) - Albert Chan - 12-04-2019 03:03 PM

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