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drew
2003-11-24 14:31:01
Herons formula

Out of curiosity has anyone derived herons formula?

SilverKnight
2003-11-24 14:35:48
Re: Herons formula

yeah.... Heron did.

SilverKnight
2003-11-24 14:42:09
Re: Herons formula

You can find a proof of Heron's formula at:
http://mathforum.org/library/drmath/view/54957.html, and
http://planetmath.org/encyclopedia/ProofOfHeronsFormula.html

Victor Zapana
2003-11-24 17:01:30
Re: Herons formula

smary answer sk... Heron did... well i have the paper of deriving it... but im too lazy to scan it or type it up here lol

Victor Zapana
2003-11-24 17:01:47
Re: Herons formula

hmm.. i cant spell (to above post) smart*

Victor Zapana
2003-11-24 17:03:56
Re: Herons formula

Can someone derive Stewart's Theorem for me? i forgot how to... just for those who don't know what i mean its the MAN + DAD = BMB + CNC with the cevian and the triangle... sorry for being so nondescriptive.

drew
2003-11-24 20:16:48
Re: Herons formula

another question for curiosity was heron the guys name or what?

drew
2003-11-24 20:20:32
Re: Herons formula

to silver knight those proofs are mean anyone got some that does not use the law of cosines?

drew
2003-11-24 20:23:15
Re: Herons formula

oh yah i forgot to ask if anyone has created a program that factors equations for a graphing calculator. No i am not in algebra 2........ i wont cheat

Victor Zapana
2003-11-24 21:07:24
Re: Herons formula

hold on til tomorrow drew i have the paper in my locker in my school. it has the derivation to the herons formula using no trig rule. so wait. patience is a virture

drew
2003-11-24 21:56:58
Re: Herons formula

ok thnx i take it my other questions will not be awnsered?

Victor Zapana
2003-11-26 17:13:13
Re: Herons formula

hmm ok.. sry for doing this so late. yesterday i was slightly ill.. so i dint have the strength to do it yesterday but today is fine.


There is a triangle, with sides a, b, c. The altitude h is made so that a is the base. Altitude h splits a into parts p and q. c is to the left of the altitude, and so it p. b is the right of the altitude, and so is q. Thats the triangle and all the sides needed to prove Herons formula


h= c^2 - p^2 = b^2 - q^2 (important equation needed to be used later)


- p^2 = b^2 - q^2 - c^2


q^2 - p^2 = b^2 - c^2 = (q + p)(q - p)


q + p = a (because p and q are the 2 parts that make up side a)


b^2 - c^2 = a(q - p)= aq - ap (now leave alone for just a second)


a^2 = a(p + q) = ap + aq


a^2 + b^2 - c^2 = 2aq (important equation needed to be used later)


(Area of Triangle) K= (1/2)(a)(h)


K^2 = (1/4)(a^2)(h^2)


16K^2 = (16)(1/4)(a^2)(h^2)


= 4(a^2)(h^2)


= 4(a^2)(b^2 - q^2) (this is from above equation)


= 4(a^2)(b^2) - 4(a^2)(q^2)


= (2ab + 2aq) (2ab - 2aq)


= (2ab + a^2 + b^2 - c^2) (2ab - a^2 - b^2 + c^2) (from above equation)


= (a^2 + 2ab + b^2 - c^2) (c^2 - a^2 + 2ab - b^2)


= ((a + b)^2 - c^2) (c^2 - (a - b)^2) (Now factor)


16K^2 = (a + b + c)(a + b - c)(c + a - b)(c - a + b)


K^2 = [(a + b + c)(a + b - c)(c + a - b)(c - a + b)]/16


K^2 = [(a + b + c)/2] [(a + b - c)/2] [(c + a - b)/2] [(c - a + b)/2]


Lets say s = [(a + b + c)/2] (semi-perimeter)


K^2 = (s)(s - c)(s - b)(s - a)


K = sqrt[s(s - c)(s - b)(s - a)]


There ya go. enjoy :)

Victor Zapana
2003-11-27 11:20:41
Re: Herons formula

er my bad the first equation is h^2 = blah blah blah, not h = blah blah blah, now the proof makes sense

drew
2003-11-27 11:24:05
Re: Herons formula

could you make it any longer?

Victor Zapana
2003-11-27 23:25:06
Re: Herons formula

lol... u wanted it without trig so here. it cant be shortened me thinks

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